Physics for Engineering

J. Gordon Leishman

5 Physics for Engineering

Introduction

“Leges naturae per physicam revelantur.“^[1] The motion of aircraft, the flow of air, the production of thrust, the transmission of structural loads, and the behavior of spacecraft in orbit are all governed by the same fundamental physical laws. Physics is about observing how the world behaves and asking why it behaves that way. Mathematics supplies the formal language of engineering, but physics provides the interpretation that ties those equations to real flight vehicles. Without that grounding, mathematical equations are just symbols to be manipulated rather than quantitative descriptions of any physical behavior.

Physics^[2] enables engineers to model matter and energy, formulate governing equations, and predict how all types of systems respond over a wide range of conditions. The laws of motion, conservation of energy and momentum, gravitation, and electromagnetism form the foundation of all aerospace analysis, applying equally to an aircraft accelerating along a runway and to a satellite in orbit. With experience, engineers learn to assign precise physical meanings to each term in the relevant governing equations and to use mathematics as a tool for deriving understanding and obtaining quantitative results.

Most readers of this eBook will have taken introductory physics courses. Still, a focused review of the core principles of mechanics is essential before studying aerodynamics, propulsion, structures, flight dynamics, and orbital mechanics. The theory and understanding of flight demands an integrated application of physics because it involves coupled translational and rotational motion, significant variations in velocity, thermodynamics, and strong interactions among forces, energy, and momentum.

A central skill in engineering is the construction of a correct free-body diagram. A free-body diagram isolates a body or system and identifies all external forces and moments acting on it; only after this physical model is established can the relevant governing equations be written correctly. Engineers who learn to draw accurate free-body diagrams and systematically apply Newton’s and other physical laws will find that complex aerospace problems become much more tractable. Physica sine mathematica caeca est, mathematica sine physica inanis est.^[3]

Learning Objectives

Review Newton’s laws of motion and their applications.
Recall the concepts of free-body diagrams, force, mass, acceleration, and inertial reference frames.
Summarize the principles of work and energy to solve engineering problems.
Review how to use the methods of linear momentum and impulse.
Understand the fundamentals of rotational motion and angular dynamics.
Revisit Newton’s law of gravitation and its application to orbital motion.
Summarize the basic principles of electricity and magnetism relevant to aerospace systems.
Describe the basis of optics and ray tracing.
Review the principles of relativity.

Physical Laws in Engineering

Engineering analysis proceeds by expressing the physical laws in mathematical form and applying them consistently to a clearly defined system. In this eBook, various types and sets of governing equations are derived directly from the conservation laws of mass, momentum, energy, and angular momentum, with careful attention to assumptions, reference frames, and boundary conditions. These equations then become quantitative statements of physical behavior. Readers seeking a deeper conceptual perspective on the foundations of physics are encouraged to consult the Feynman Lectures, which remain one of the most insightful expositions of the subject.

Conservation Principles

Conservation principles in physics are fundamental laws that describe the conservation of specific quantities in physical systems. These laws state that specific physical quantities will remain constant in isolated systems or undergo specific transformations under certain conditions.

1. Conservation of Mass (Conservatio Massae):

“Mass is neither created nor destroyed in a closed system; it remains constant.”

This principle states that the total mass of a closed system remains unchanged in any physical or chemical process, i.e.,

(1) $\begin{equation*} \sum_{\mbox{\tiny Something}} m_{\rm after} = \sum_{\mbox{\tiny Something}} m_{\rm before} \end{equation*}$

where $m$ denotes the mass of something.

2. Conservation of Momentum (Conservatio Momentum):

“The total momentum of an isolated system remains constant if no external forces act upon it.“

Momentum, defined as the product of an object’s mass and velocity, is conserved in a closed system where no external forces are present, i.e.,

(2) $\begin{equation*} \sum_{\mbox{\tiny Something}} p_{\rm after} = \sum_{\mbox{\tiny Something}} p_{\rm before} \end{equation*}$

where $p$ denotes linear or translational momentum.

3. Conservation of Angular Momentum (Conservatio Momenti Angulares):

“The total angular momentum of an isolated system remains constant without external torques.”

Angular momentum, which is related to an object’s rotational motion, is conserved when no external torques are applied to the system, i.e.,

(3) $\begin{equation*} \sum_{\mbox{\tiny Something}} l_{\rm after} = \sum_{\mbox{\tiny Something}} l_{\rm before} \end{equation*}$

where $l$ denotes angular momentum.

4. Conservation of Energy (Conservatio Energiae):

“Energy cannot be created or destroyed; it can only be converted from one form to another or transferred between different objects or systems.”

This principle states that the total energy of an isolated system remains constant over time, i.e.,

(4) $\begin{equation*} \sum_{\mbox{\tiny Something}} e_{\rm after} = \sum_{\mbox{\tiny Something}} e_{\rm before} \end{equation*}$

where $e$ denotes the energy.

Fundamental physical laws for a rigid body, showing the state variables of the control mass bounded by the conservation of mass, momentum, angular momentum, and energy.

Newton’s Absolutes

Isaac Newton’s absolutes refer to the fundamental concepts introduced in his formulation of classical mechanics, specifically in his book Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy). These concepts include absolute space, absolute time, and absolute motion.

1. Absolute time: According to Newton, absolute time flows uniformly and continuously, independent of any events or objects. It is not influenced by the physical processes occurring within it. This concept implies a universal time that is the same everywhere, providing a consistent measure of an event’s duration.

2. Absolute space: Newton posited the existence of a fixed, immovable, and infinite space that exists independently of any objects within it. This space provides a reference frame for measuring the position and motion of objects. Absolute space remains constant and unchanging, serving as a basis for describing all physical events.

3. Absolute motion: Newton also distinguished between absolute motion and relative motion. Absolute motion refers to the movement of an object through absolute space, while relative motion is the movement of an object relative to other objects. Absolute motion is measured with respect to the fixed framework of absolute space. Newton’s absolutes were foundational to the development of his laws of motion and the effects of gravity. However, Einstein’s theory of relativity later challenged and refined these ideas, demonstrating that space and time are not absolute but interconnected and relative to the observer.

Newton’s Laws

Isaac Newton formulated three fundamental laws of motion in his Principia. These laws laid the foundation for classical mechanics and remain essential for understanding the scientific principles underlying many fields of science and engineering. Although Newton’s laws of motion now appear intuitive and almost self-evident to many, at the time of their formulation, they represented a profound paradigm shift in the understanding of nature. They replaced centuries of qualitative reasoning with a rigorous quantitative framework that unified force, mass, and motion into a single formalism. This framework enabled the systematic development of engineering science and remains central to modern physics and engineering analysis.

1. Newton’s First Law of Motion (Law of Inertia). Lex prima motus Newtonii (Lex inertiae):

“An object at rest tends to stay at rest, and an object in motion tends to stay in motion with the same speed and in the same direction unless acted upon by an external force.”

Inertia is the property of an object that resists changes in its motion. According to the first law, an object will maintain its state of rest or uniform motion in a straight line unless a net external force acts on it. This law establishes the existence of inertial reference frames and emphasizes that force is required only to change motion, not to sustain it.

2. Newton’s Second Law of Motion (Law of Acceleration). Lex secunda motus Newtonii (Lex accelerationis):

“The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.”

This law quantifies the relationship between force, mass, and acceleration and provides the basis for all dynamical analysis. It is most commonly expressed mathematically as

(5) $\begin{equation*} F = m \, a \end{equation*}$

where $m$ is the mass, $F$ is the force, and $a$ is the acceleration, or, in its more fundamental vector and momentum form, as

(6) $\begin{equation*} \vec{F} = \frac{d}{dt} ( m \, \vec{V} ) = \frac{d \vec{p}}{dt} \end{equation*}$

where $\vec{p} = m \, \vec{V}$ is the linear momentum of the body. Although conservation of angular momentum is implicit in the Principia, particularly through the law of areas for central forces, angular momentum was not identified or treated as a fundamental dynamical quantity by Newton in his Principia.

3. Newton’s Third Law of Motion (Law of Action-Reaction). Lex tertia motus Newtonii (Lex actionis et reactionis):

“For every action, there is an equal and opposite reaction.”

According to the third law, whenever one body exerts a force on another body, the second body simultaneously exerts an equal and opposite force on the first body, i.e.,

(7) $\begin{equation*} \vec{R} = - \vec{F} \end{equation*}$

This law governs all mechanical interactions and emphasizes that forces always occur in action-reaction pairs, equal in magnitude and opposite in direction, acting on different bodies.

Newton’s laws of motion have profoundly influenced the development of science and engineering. They have provided a systematic framework for understanding the behavior of objects in motion and at rest, thereby advancing the fields of mechanics, aeronautics, astronautics, astronomy, and other branches of engineering, physics, and thermodynamics.

In engineering analysis, Newton’s laws are seldom applied in their differential form alone. Instead, they are most often used in accordance with the associated conservation principles of mass, linear momentum, angular momentum, and energy, applied to a finite body or control mass. Nevertheless, these conservation statements provide the operational form in which Newton’s laws are used to model a wide range of physical systems.

Force, Mass, & Acceleration

The fundamental quantities that appear in Newtonian mechanics are force $vec{F}$ , mass $m$ , and acceleration $\vec{a}$ . These quantities provide the physical content of the conservation principles and the laws of motion. Their proper interpretation is inseparable from the choice of reference frame in which motion is observed and measured. The mathematical statements of classical mechanics are predictive only when these concepts are clearly established.

A force represents a physical interaction between bodies that is capable of altering the state of motion of a system. In classical mechanics, force is treated as a vector quantity and so possesses both magnitude and direction. The dynamical role of force is expressed through Newton’s second law of motion, i.e.,

(8) $\begin{equation*} \sum \vec{F} = m \, \vec{a} \end{equation*}$

This relationship makes clear that a net force does not create velocity but produces an acceleration, that is, a change in velocity. A body may move with constant velocity in the complete absence of force. Only when the velocity is changing in either magnitude or direction does a net force exist.

The quantity of mass is a measure of a body’s resistance to changes in its state of motion, whether by changes in speed or direction, and is, therefore, a measure of inertia. Inertial mass is defined operationally through Newton’s second law, which has already been explained, i.e.,

(9) $\begin{equation*} \sum \vec{F} = m \, \vec{a} \end{equation*}$

which identifies mass as the proportionality constant between the net applied force and the resulting acceleration. Mass is an intrinsic property of matter and is independent of the body’s location, orientation, or surrounding environment.

Mass must be carefully distinguished from weight, which is the gravitational force exerted on a mass by a nearby gravitating body. Near the surface of the Earth, the weight vector is

(10) $\begin{equation*} \vec{W} = m \, \vec{g} \end{equation*}$

where $\vec{g}$ is the local gravitational field vector. The magnitude of the weight is, therefore, $W = m \, g$ . Unlike mass, weight depends on both position and the gravitational environment. The symbol ${g_0}$ is often used to denote the value of $g$ at mean sea level (MSL), particularly in the analysis of spacecraft. Because mass is constant while weight varies with location in the universe, the two quantities must not be confused.

Mass and weight are distinct physical quantities. Mass measures inertia, whereas weight is the gravitational force acting on that mass.

Acceleration is defined kinematically as the time rate of change of velocity, i.e.,

(11) $\begin{equation*} \vec{a} = \frac{d \vec{V}}{dt} \end{equation*}$

Because velocity is a vector quantity, acceleration reflects not only changes in speed but also changes in the direction of motion. Consequently, a body undergoing circular motion at constant speed experiences a nonzero acceleration even though the magnitude of its velocity remains unchanged. In Newtonian mechanics, acceleration is the immediate dynamical response of a body to a net external force.

Remember that speed is a scalar quantity that describes how fast a body is moving and is independent of direction. It is the magnitude of the velocity vector and is always nonnegative. Velocity, by contrast, is a vector quantity that specifies both the rate of motion and the direction of motion. Because velocity includes direction, it can change even when speed remains constant.

Reference Frames & Coordinate Transformations

The interpretation of force, mass, and acceleration is inseparable from the concept of a reference frame. A reference frame is the coordinate system from which motion is observed and measured. All statements of position, velocity, and acceleration are meaningful only when referred to a specified reference frame. Newton’s laws are valid only in inertial reference frames, which are defined as frames in which a body subject to zero net external force moves with constant velocity. Any frame translating at constant velocity relative to an inertial frame is itself inertial. In such frames, Eq. 9 retains its simplest and most fundamental form.

Reference frames that accelerate or rotate relative to an inertial frame are called non-inertial frames. When equations of motion are written in a non-inertial frame, Newton’s laws no longer apply in their standard form because the reference frame itself undergoes acceleration or rotation. To preserve the familiar structure of Newton’s second law, additional apparent or fictitious forces must be introduced. These include the centrifugal force that is associated with rotation, the Coriolis force associated with motion relative to a rotating frame, and the Euler force associated with time-varying rotation rates. These forces account for the acceleration of the reference frame itself and allow the equations of motion to be written in a form analogous to those used in inertial frames. When properly included, they permit the consistent application of Newtonian mechanics in accelerating and rotating coordinate systems.

Accordingly, engineering dynamics begins not with equations but with the specification of a reference frame and the identification of all fundamental external forces acting on the chosen system. This goal is accomplished by defining a control mass and constructing its corresponding free-body diagram, thereby establishing the system’s physical boundary or boundaries, and the forces and moments acting on it. Only after this physical model is complete can the governing equations be written correctly.

Translating Reference Frames

Let $O$ be an inertial reference frame and $O'$ a reference frame whose origin translates with position $\vec{R}(t)$ relative to $O$ , as shown in the figure below. Quantities with primes denote particle motion measured in the non-inertial frame, while unprimed frame quantities describe the motion of the reference frame itself. The position vectors of a particle measured in the two frames satisfy the purely geometric relationship

(12) $\begin{equation*} \vec{r}(t) = \vec{r}^{\,\prime}(t) + \vec{R}(t) \end{equation*}$

Differentiation with respect to time gives the corresponding velocity and acceleration transformations, i.e.,

(13) $\begin{equation*} \vec{v}(t) = \vec{v}^{\,\prime}(t) + \vec{V}(t) \qquad \text{and} \qquad \vec{a}(t) = \vec{a}^{\,\prime}(t) + \vec{A}(t) \end{equation*}$

where $\vec{V}(t) = \overbigdot{\vec{R}}(t)$ and $\vec{A}(t) = \overbigddot{\vec{R}}(t)$ are the velocity and acceleration of the translating frame $O'$ relative to $O$ .

Schematic illustration of translating reference frames.

Newton’s second law, written in the inertial frame $O$ based on the sum of the physical forces, is

(14) $\begin{equation*} \sum \vec{F}_{\text{physical}} = m\,\vec{a} \end{equation*}$

Substituting the acceleration transformation in Eq. 13 into this expression gives

(15) $\begin{equation*} m\,\vec{a}^{\,\prime} = \sum \vec{F}_{\text{physical}} - m\,\vec{A} \end{equation*}$

The additional term in this equation, which is proportional to the frame acceleration, arises solely because the equations of motion are written in an accelerating reference frame. Therefore, this term is identified as a fictitious, or inertial, force, i.e.,

(16) $\begin{equation*} \vec{F}_{\text{fictitious}} = -m\,\vec{A} \end{equation*}$

With this definition, Newton’s second law in the accelerating frame $O'$ can be written in the familiar form

(17) $\begin{equation*} \sum \vec{F}_{\text{physical}} + \vec{F}_{\text{fictitious}} = m\,\vec{a}^{\,\prime} \end{equation*}$

As an example, consider an aircraft accelerating forward along the runway during takeoff with acceleration $\vec{A} = A\,\hat{x}$ . In the aircraft-fixed frame $O'$ , a mass inside the aircraft experiences a fictitious force $-m\, A\,\hat{x}$ acting opposite the direction of the acceleration. This force accounts for the sensation of being pushed backward into the seat and must be included in force balances written in the accelerating aircraft frame, even though no additional physical interactions are present. Therefore, Newton’s second law in the accelerating frame $O'$ can be written in the familiar form as

$\sum \vec{F}_{\text{physical}}+\vec{F}_{\text{fictitious}}=m\,\vec{a}^{\,\prime}$

Rotating Reference Frames

Consider a reference frame $O'$ that rotates with angular velocity $\vec{\Omega}(t)$ relative to an inertial frame $O$ , as shown in the figure below, with both frames sharing a common origin so that the position vector $\vec{r}$ is the same in both frames. The time derivative of the position vector measured in the rotating frame is related to its inertial derivative by

(18) $\begin{equation*} \left( \frac{d\vec{r}}{dt} \right)_O = \left( \frac{d\vec{r}}{dt} \right)_{O'} + \vec{\Omega} \times \vec{r} \end{equation*}$

the cross-product giving another velocity vector.

Schematic illustration of a rotating reference frame.

Applying this relationship to the position vector $\vec{r}$ gives the velocity transformation

(19) $\begin{equation*} \vec{v} = \vec{v}^{\,\prime} + \vec{\Omega} \times \vec{r} \end{equation*}$

Differentiating once more gives the acceleration transformation, i.e.,

(20) $\begin{equation*} \vec{a} = \vec{a}^{\,\prime} + 2\,\vec{\Omega} \times \vec{v}^{\,\prime} + \vec{\Omega} \times (\vec{\Omega} \times \vec{r}) + \overbigdot{\vec{\Omega}} \times \vec{r} \end{equation*}$

where $\vec{v}^{\,\prime}$ and $\vec{a}^{\,\prime}$ are the velocity and acceleration measured in the rotating frame, and $\overbigdot{\vec{\Omega}}$ is the angular acceleration of the frame. Newton’s second law, written in the inertial frame, is

(21) $\begin{equation*} \sum \vec{F}_{\text{physical}} = m\,\vec{a} \end{equation*}$

Substituting the acceleration transformation and rearranging terms gives

(22) $\begin{equation*} m\,\vec{a}^{\,\prime} = \sum \vec{F}_{\text{physical}} - m\left[ 2\,\vec{\Omega} \times \vec{v}^{\,\prime} + \vec{\Omega} \times (\vec{\Omega} \times \vec{r}) + \overbigdot{\vec{\Omega}} \times \vec{r} \right] \end{equation*}$

The additional terms here arise solely because the equations of motion are written in a rotating reference frame. Therefore, they are identified as fictitious or inertial forces. With these definitions, Newton’s second law in the rotating frame may be written in the familiar form as

(23) $\begin{equation*} \sum \vec{F}_{\text{physical}} + \vec{F}_{\text{Coriolis}} + \vec{F}_{\text{centrifugal}} + \vec{F}_{\text{Euler}} = m\,\vec{a}^{\,\prime} \end{equation*}$

where

(24) $\begin{equation*} \vec{F}_{\text{Coriolis}} = -2m\,\vec{\Omega} \times \vec{v}^{\,\prime} \ , \quad \vec{F}_{\text{centrifugal}} = -m\,\vec{\Omega} \times (\vec{\Omega} \times \vec{r}) \ , \quad \text{and} \quad \vec{F}_{\text{Euler}} = -m\,\overbigdot{\vec{\Omega}} \times \vec{r} \end{equation*}$

The centrifugal force depends only on position within the rotating frame and is present even when the particle is stationary relative to that frame. The Coriolis force depends on the velocity relative to the rotating frame and acts perpendicular to the direction of motion. The Euler force arises only when the rotation rate of the frame is time-varying. These three forces appear in many engineering problems and account for the kinematics of the rotating reference frame itself, and then allow Newton’s laws to be applied to rotating coordinate systems.

The Free-Body Diagram

The free-body diagram (FBD) is the primary conceptual tool that connects a physical system to its mathematical description. It is the starting point for almost every dynamics, vibration, structural, and fluid-mechanical analysis in engineering. An equation written without reference to a correct FBD has limited physical meaning, regardless of how sophisticated the mathematics may appear. From the perspective of conservation laws, the FBD defines the system boundary across which linear momentum, angular momentum, and energy are conserved. It is not just a graphical aid but the formal physical specification of the problem to which Newton’s laws are applied in conservation form.

When Newton’s laws are applied in conservation form, they are interpreted as balance statements written for a specified system rather than as local kinematic identities. For a control mass defined by the free-body diagram, the laws express that the time rate of change of a conserved quantity inside the system equals the net flux of that quantity across the system boundary plus any external sources. For linear momentum, this balance is written as

(25) $\begin{equation*} \sum \vec{F}_{\text{ext}} = \frac{d}{dt} \left( m \, \vec{V}_{\mathrm{cm}} \right) \end{equation*}$

where the left-hand side represents all forces acting across the system boundary and the right-hand side represents the rate of change of linear momentum of the control mass. An analogous balance applies to angular momentum and to energy. Therefore, the free-body diagram specifies not only which forces and moments appear in the equations of motion but also which system the conservation laws apply to. Without that specification, the meaning of the resulting equations is ambiguous.

The construction of a FBD begins by isolating the body (control mass) from its surroundings. Every physical interaction between the body and its environment must then be shown explicitly as an external force or an external moment acting on the isolated body. Each force is drawn with a definite direction, point of application, and physical origin. Common examples include gravitational, contact, pressure, friction, and various types of forces and moments. External moments can be produced by applied couples, by forces acting with an offset line of action, by distributed loads such as aerodynamic pressure, and by reactions at supports. Simple examples are shown in the figure below, in which the net forces on each vehicle are resolved at its center of gravity.

Simple examples of free-body diagrams where the forces are resolved to the center of gravity. The governing translational equations follow directly from the force balances along orthogonal directions.

Only interactions that cross the chosen system boundary should be included on a free-body diagram. Forces exchanged between parts of the same isolated system are internal; they occur in equal and opposite pairs and so do not influence the motion of the system as a whole. Including internal forces or internal couples in a free-body diagram is a conceptual error that almost always leads to incorrect forms of the governing equations.

Once the FBD is constructed, Newton’s laws provide the direct link between the physical interactions and the resulting motion. For the translational motion of a particle or a rigid body in Cartesian coordinates, then

(26) $\begin{equation*} \sum F_x = m \, a_x \, , \qquad \sum F_y = m \, a_y \, , \quad \text{and} \quad \sum F_z = m \, a_z \end{equation*}$

which is equivalent to the vector balance

(27) $\begin{equation*} \sum \vec{F} = m \, \vec{a} \end{equation*}$

The acceleration in these equations is the acceleration of the center of mass in the chosen inertial reference frame.

For rigid bodies, a translational balance is necessary but insufficient. The angular momentum balance governs the rotational motion, i.e.,

(28) $\begin{equation*} \sum \vec{M}_O = \frac{d \vec{L}_O}{dt} \end{equation*}$

where $\sum \vec{M}_O$ is the resultant of all external moments about a reference point $O$ , and $\vec{L}_O$ is the angular momentum of the body about that point. Therefore, a complete rigid-body FBD must contain all external forces and any applied couples needed to support both the force balance and the moment balance.

To this end, an elaboration of the helicopter example given above can now be developed. Assume that the rotor thrust and the drag act at fixed, known points of application, as shown in the figure below. Aerodynamic drag acts at an effective center of pressure. The rotor thrust $T$ , which can be assumed to act at the hub and is perpendicular to the plane of rotor rotation, is inclined by the disk tilt angle $\alpha$ , so that its vertical component of thrust balances the weight and its horizontal component balances the drag, i.e.,

(29) $\begin{equation*} T \cos \alpha = W \qquad \text{and} \qquad T \sin \alpha = D \end{equation*}$

Because the lines of action of $T$ and $D$ do not intersect any one reference point, in this case, the center of gravity, the force balance alone is insufficient. A simultaneous moment balance is also required, which, according to the figure, gives in this case

(30) $\begin{equation*} T\cos\alpha\, l_1 = D\, l_2 \end{equation*}$

In steady trim, these three equations must be satisfied simultaneously. The two force-balance equations can be solved directly for the thrust magnitude and tilt angle as

(31) $\begin{equation*} \tan\alpha = \frac{D}{W} \qquad \text{and} \qquad T = \sqrt{W^2 + D^2} \end{equation*}$

Substituting $T\cos\alpha = W$ into the moment equation shows that

(32) $\begin{equation*} W \, l_1 = D\, l_2 \end{equation*}$

which is the trim condition that must be satisfied by the geometry ( $l_1$ , $l_2$ ) and the forces $W$ and $D$ . If this condition is not met, simultaneous force and moment equilibrium is impossible for the assumed configuration.

Free-body diagram of a helicopter in steady forward flight illustrating simultaneous force and moment equilibrium.

Check Your Understanding #1 – Newton’s Laws of Motion

A block of mass $m$ = 12 kg rests on a rough horizontal surface. The coefficient of kinetic friction is $\mu_k$ = 0.25. A force of magnitude $F$ = 60 N is applied at an angle $\theta$ = 30 $^\circ$ above the horizontal. Draw a free-body diagram (FBD) of the problem, then determine: (a) the acceleration of the block, and (b) the time required for the block to reach a speed of 8 m/s, assuming it starts from rest.

Show solution/hide solution.

The applied force is resolved into horizontal and vertical components, as shown in the FBD.

The equations of motion in the horizontal and vertical directions are

$\sum F_x = m\,a \quad \text{and} \quad \sum F_y = 0$

with

$F_x = F\,\cos\theta \qquad \text{and} \qquad F_y = F\,\sin\theta$

The vertical force balance gives the normal force $N$ as

$N = m\,g - F\,\sin\theta$

The kinetic friction force is

$F_\mu = \mu_k\,N = \mu_k \left( m\,g - F\,\sin\theta \right)$

Therefore, the horizontal equation of motion becomes

$F\,\cos\theta - F_\mu = m\, a$

Substituting numerical values gives

$60\,\cos 30^\circ - 0.25 \left( 12\,\times 9.81 - 60\,\sin 30^\circ \right) = 12\,a$

$51.96 - 21.93 = 12\,a$

so $a$ = 2.50 m/s $^2$ . Because the block starts from rest and the acceleration is constant, then $V = a\,t$ , so the time to reach $V$ = 8m/s is

$t = \frac{V}{a} = \frac{8}{2.50} = 3.2\,\text{s}$

In both particle and rigid-body dynamics, the most persistent mistakes by students and inexperienced engineers arise not from the mathematics but from the physics encoded in the free-body diagram. Common errors include omitting a force or moment, assigning an incorrect direction to a force (e.g., friction or aerodynamic drag), or introducing a linear or angular acceleration that is incompatible with the motion’s geometry. Each of these errors can often be traced directly to a flawed free-body diagram rather than to an algebraic misstep. That said, algebraic errors are human, and triple-checking the steps is always necessary to avoid such errors.

For this reason, the free-body diagram is not merely a sketch added for convenience. It is the formal physical statement of the problem. Once written down, it must be checked and rechecked. Once the diagram is correct and complete, the governing equations follow directly and unambiguously from Newton’s laws, the work-energy theorem, or the angular momentum balance. If the FBD is wrong, no amount of mathematical manipulation or sophistication will be able to repair the analysis. The more examples one works through using free-body diagrams, the more likely it is that errors will be minimized and proficiency will be gained.

Check Your Understanding #2 – Force and moment “trim” in flight

Consider the airplane shown in the sketch below, flying in trim with force and moment equilibrium. The airplane’s weight, $W$ , can be assumed to act at its center of gravity. The resultant wing lift, $L_W$ , and total drag, $D$ , act at the center of lift at a distance $l_1$ behind the center of gravity. The resultant tail lift, $L_T$ , acts at $l_2$ behind the center of gravity. The thrust from underslung engines acts at a distance $l_3$ below the center of gravity.

Redraw and explain the free-body diagram (FBD) of the airplane.
Write down the equations for vertical and horizontal force equilibrium.
Write down an equation for the pitching moment equilibrium about the center of gravity.
Explain the effects of varying engine thrust on the pitching moment and why this might be a concern for the flight behavior of the airplane.

Show solution/hide solution.

1. In this case, the FBD can be detailed as shown below. It need not be drawn to scale; clarity is more important.

2. In steady, unaccelerated, equilibrium flight, called “trim,” then for vertical equilibrium

$L_W - L_T - W = 0 \quad {\rm or} \quad L_W - L_T = W$

For horizontal equilibrium, then

${ T - D = 0 \quad {\rm or} \quad T = D }$

3. The airplane’s net pitching moment about its center of gravity must be zero, i.e.,

$T \, l_3 - L_W \, l_1 + L_T \, l_2 = 0$

4. An issue of concern arises from the varying engine thrust caused by underslung engines, resulting in a thrust/pitch coupling effect. For example, a pitch-up moment response will occur if engine thrust increases at the trim state. Similarly, there will be a pitch-down response for a reduction in thrust. Such effects can be compensated for (balanced) by modulating the lift on the tail, which is obtained by elevator control and/or pitch trim of the horizontal tail. You can read more about this issue here in a technical report from engineers at NASA.

Work, Energy, & Conservation of Energy

The concepts of work and energy provide an alternative but fully equivalent description of mechanical systems to that obtained from Newton’s laws. Whereas Newtonian formulations emphasize forces and accelerations, energy methods emphasize scalar measures of a system’s capacity to perform mechanical work. The equivalence of the force and energy descriptions follows directly from Newton’s second law through the work-energy theorem. Energy methods are compelling not because they are simpler, but because they remain valid for systems of great geometric and kinematic complexity and because total energy is conserved for all isolated systems. For this reason, energy principles form one of the unifying analytical frameworks of engineering physics, and many examples are given throughout this eBook.

In mechanics, work represents the transfer of mechanical energy by the action of a force through a displacement. For a particle subjected to a force $\vec{F}$ and undergoing an infinitesimal displacement $d\vec{s}$ , the differential work is defined as

(33) $\begin{equation*} dW = \vec{F} \bigcdot d\vec{s} \end{equation*}$

The total work done as the particle moves from position ${ s_1 }$ to position ${s_2}$ along an arbitrary path is

(34) $\begin{equation*} W = \int_{s_1}^{s_2} \vec{F} \bigcdot d\vec{s} \end{equation*}$

Only the component of the force parallel to the displacement contributes to the work. Forces normal to the direction of motion do no work and so do not directly alter the kinetic energy of the particle. This observation underlies the central role of constraint forces in mechanical systems.

The kinetic energy of a particle is the scalar measure of energy associated with its translational motion. For a particle of mass $m$ moving with speed $V$ , then

(35) $\begin{equation*} T = \frac{1}{2} \, m \, V^2 \end{equation*}$

Kinetic energy is nonnegative and depends on the square of the speed. Consequently, kinetic energy is far more sensitive to changes in velocity than momentum, especially for massive bodies, a distinction that becomes fundamental in high-speed aerodynamics and propulsion.

The work-energy theorem provides the exact connection between Newton’s second law and the scalar description of motion. For a particle acted upon by an arbitrary system of forces, the net work equals the change in kinetic energy, i.e.,

(36) $\begin{equation*} W_{\text{net}} = T_2 - T_1 \end{equation*}$

This relation follows directly from the vector equation of motion $\sum \vec{F} = m \, \vec{a}$ through the identity $\vec{a} \bigcdot \vec{V} = d(V^2/2)/dt$ . This theorem is not an independent physical principle but a transformed statement of Newton’s second law in energetic form. Its value lies in reducing vector dynamics to a scalar energy balance, often eliminating the need to resolve individual force components or determine the detailed time history of the motion.

If the net work done by all forces on a particle can be evaluated between two states, then the corresponding change in speed follows immediately from the change in kinetic energy. This approach is especially useful when forces vary along a path or when the time history of the motion is complicated to obtain, but the total work is easier to compute. To this end, consider a particle moving along a curved path from position 1 to position 2. At a representative point on the path, the velocity vector ${\vec{V}}$ is tangent to the trajectory, and the net force vector is $\vec{F}$ . The force can be decomposed into a tangential and a normal component. The tangential component of the force does work as the particle moves along the path and directly contributes to the change in its kinetic energy. In contrast, the normal component alters only the direction of the velocity. Therefore, the accumulated tangential work of all forces along the entire path is precisely equal to the net change in kinetic energy, i.e., $T_2 - T_1$ .

The work-energy theorem for a particle moving along a curved path from position 1 to position 2.

A force is classified as conservative if its work depends only on the initial and final positions of the particle and is independent of the path taken between them. For such forces, a scalar potential-energy function $U$ exists such that

(37) $\begin{equation*} \vec{F} = - \nabla U \end{equation*}$

The work done by a conservative force between two positions is then

(38) $\begin{equation*} W = - ( U_2 - U_1 ) \end{equation*}$

Uniform gravitational, inverse-square, and linear-elastic spring forces are conservative. Friction, aerodynamic drag, and electrical resistance are nonconservative because their work depends on the detailed path of motion, resulting in irreversible loss of mechanical energy.

The mechanical energy of a particle is defined as the sum of its kinetic and potential energies, i.e.,

(39) $\begin{equation*} E = T + U \end{equation*}$

This quantity represents the total recoverable energy associated with motion and configuration in a conservative force field. For a particle acted upon only by conservative forces, then

(40) $\begin{equation*} T_1 + U_1 = T_2 + U_2 \end{equation*}$

Mechanical energy is conserved and continuously interchanges between kinetic and potential forms without loss. When nonconservative forces act, the change in mechanical energy is equal to the work performed by those forces. It provides a measure of energy dissipation, i.e., an irrecoverable energy loss.

The concept of power comes up in many engineering applications, and is defined as the time rate at which work is done, or energy is transferred, i.e.,

(41) $\begin{equation*} P = \frac{dW}{dt} \end{equation*}$

Using the definition of work, the instantaneous mechanical power associated with a force acting on a particle moving with velocity ${\vec{V} }$ is

(42) $\begin{equation*} P = \vec{F} \bigcdot \vec{V} \end{equation*}$

which is equivalent to a force times a velocity.

In practical engineering systems, power is the governing quantity that determines instantaneous performance. The thrust of a propeller, the hovering flight capability of a helicopter, the climb performance of an aircraft, the acceleration of a vehicle, and the output of an electric motor are all ultimately limited by available power rather than by force or energy alone. Structural limits cap allowable forces and energy capacity, whereas power fixes the instantaneous operating envelope of the system or machine.

The work-energy theorem, conservation of mechanical energy, and power relations in various forms will recur throughout the analysis of propulsion systems, rotating machinery, flight performance, and energy-conversion devices. Together with Newton’s laws and the conservation principles, they form one of the central analytical pillars of engineering physics.

Check Your Understanding #3 – Work & Energy

A main landing gear oleo-pneumatic strut can be idealized as a linear spring of stiffness $k$ = 2.0 $\times10^{5}$ N/m in parallel with a hydraulic damper In practice, oleo struts are metered so that the hydraulic resisting force is approximately load-limited and only weakly dependent on velocity over the working stroke, so assume the damper provides an approximately constant resisting force $F_d$ = 5,000 N over the stroke. The strut supports an effective mass of $m$ = 1,200 kg. At touchdown, the strut just contacts the stop (zero stroke), and the mass has a vertical sink speed of $V_0$ = 2.0 m/s downward. Assuming purely vertical motion, determine the strut’s maximum compression.

Show solution/hide solution.

Take downward as the positive direction. The forces doing work during compression are the weight, spring force, and damper force. The work-energy theorem gives

$W_{\text{net}} = T_2 - T_1$

At the instant of touchdown (zero stroke), then

$T_1 = \frac{1}{2} \, m \, V_0^2$

and at maximum compression, the mass is momentarily at rest, so $T_2 = 0$ . The weight $m g$ acts downward and does positive work over a compression distance $x$ , so

$W_g = m \, g\, x$

The spring force $k \, x$ acts upward and does negative work, so

$W_s = -\int_0^x k \, x'\,dx' = -\frac{1}{2} \, k \, x^2$

The damper force $F_d$ also acts upward, so its work is

$W_d = -F_d\, x$

Therefore, the net work is

$W_{\text{net}} = W_g + W_s + W_d = m \, g\, x - \dfrac{1}{2} \, k \, x^2 - F_d\, x$

From the work-energy theorem, then

$W_{\text{net}} = T_2 - T_1 = -\frac{1}{2} \, m \, V_0^2$

so

$m \, g\, x - \dfrac{1}{2} \, k \, x^2 - F_d\, x = -\dfrac{1}{2} \, m \, V_0^2$

Rearranging gives a quadratic equation, i.e.,

$\frac{1}{2} \, k \, x^2 - (m \, g - F_d)\, x - \frac{1}{2} \, m \, V_0^2 = 0$

Substituting the numerical values gives

$\frac{1}{2} (2.0\times10^{5}) x^2 - \left( 1{,}200(9.81) - 5{,}000\right) x - \frac{1}{2} (1{,}200)(2.0)^2 = 0$

$100{,}000\,x^2 - 6{,}772\,x - 2{,}400 = 0$

Solving this quadratic for the physically relevant (positive) root gives $x \approx$ 0.192 m.

Linear Momentum & Impulse

Whereas energy methods emphasize scalar measures of motion, momentum methods emphasize vector quantities and are particularly well suited to the analysis of interactions, impacts, and particle systems. Momentum principles arise directly from Newton’s second and third laws and lead naturally to powerful conservation results. Together with work and energy, momentum forms one of the fundamental pillars of classical mechanics used throughout engineering.

Linear momentum and impulse provide a vector-based framework for analyzing motion that complements the scalar energy methods developed in the previous section. The linear momentum of a particle is defined as the product of its mass and velocity. For a particle of mass $m$ moving with velocity ${\vec{V} }$ , the linear momentum $\vec{p}$ is

(43) $\begin{equation*} \vec{p} = m \, \vec{V} \end{equation*}$

Momentum is a vector quantity and so possesses both magnitude and direction. For a given speed, a particle with a larger mass possesses greater momentum and is correspondingly more difficult to bring to rest or to deflect from its path.

Newton’s second law is most fundamentally expressed in terms of momentum rather than acceleration. In vector form, then

(44) $\begin{equation*} \sum \vec{F} = \frac{d\vec{p}}{dt} \end{equation*}$

For a particle of constant mass, this equation reduces to

(45) $\begin{equation*} \sum \vec{F} = m \, \frac{d\vec{V}}{dt} \end{equation*}$

For systems in which the mass varies with time, the momentum form remains valid and must be used with an explicit description of mass exchange, i.e.,

(46) $\begin{equation*} \sum \vec{F} = \frac{d}{dt} \left( m \, \vec{V} \right) \end{equation*}$

This general form is the starting point for all variable-mass problems in mechanics.

Impulse-Momentum Theorem

The impulse of a force is defined as the time integral of the force over a specified time interval. If a force $\vec{F}(t)$ acts on a particle between times $t_1$ and $t_2$ , the impulse $\vec{J}$ is

(47) $\begin{equation*} \vec{J} = \int_{t_1}^{t_2} \vec{F}(t) \, dt \end{equation*}$

Impulse is a vector quantity with units of momentum and represents the cumulative effect of a force acting over time, irrespective of how the force varies within the interval.

Impulse represents the cumulative effect of a force acting over time and is equal to the corresponding change in momentum

By integrating Newton’s second law in momentum form over time, the impulse-momentum theorem follows, i.e.,

(48) $\begin{equation*} \int_{t_1}^{t_2} \sum \vec{F}(t) \, dt = \vec{p}_2 - \vec{p}_1 \end{equation*}$

or equivalently

(49) $\begin{equation*} \vec{J}_{\text{net}} = \Delta \vec{p} \end{equation*}$

This theorem states that the net impulse of all forces acting on a particle equals the change in its linear momentum. It is entirely equivalent to Newton’s second law but is often more convenient when forces act over short time intervals or vary rapidly with time.

Conservation of Linear Momentum

For an isolated system of particles, the net external force vanishes, i.e.,

(50) $\begin{equation*} \sum \vec{F}_{\text{ext}} = \vec{0} \end{equation*}$

and so

(51) $\begin{equation*} \frac{d\vec{p}_{\text{total}}}{dt} = \vec{0} \end{equation*}$

It follows that the total linear momentum of the system remains constant, i.e.,

(52) $\begin{equation*} \vec{p}_{\text{total}} = \text{constant} \end{equation*}$

This result is a direct consequence of Newton’s third law, which ensures that internal forces between particles occur in equal and opposite pairs and so cancel in the momentum balance for the system as a whole.

Check Your Understanding #4 – Linear Momentum

A mass of $m$ = 2 kg rests on a smooth horizontal air track. A horizontal force $F(t)$ is applied over a short interval and then removed. The force varies with time according to

$F(t) = 50\,t^2 \quad \text{for} \quad 0 \le t \le 0.80\,\text{s}$

with $F(t)$ = 0 outside this interval. Using the impulse-momentum theorem, determine: (a) the impulse delivered to the mass over $0 \le t \le 0.80$ s and (b) the speed of the mass at $t$ = 0.80 s. The mass is initially at rest.

Show solution/hide solution.

The impulse of a time-varying force is obtained by integrating the force over the time interval of interest, i.e.,

$J = \int_{t_1}^{t_2} F(t)\,dt$

Here $t_1 = 0$ and $t_2 = 0.80\,\text{s}$ , so

$J = \int_{0}^{0.80} 50\,t^2\,dt$

The integral is straightforward, i.e.,

$J = 50 \int_{0}^{0.80} t^2\,dt = 50 \left[ \frac{t^3}{3} \right]_{0}^{0.80} = 50 \left( \frac{0.80^3}{3} \right) = 50 \left( \frac{0.512}{3} \right) \approx 50 \times 0.1707 \approx 8.53\,\text{N}\,\text{s}$

The impulse-momentum theorem states that the net impulse equals the change in linear momentum, i.e.,

$J = \Delta p = m\,V_2 - m\,V_1$

The mass starts from rest, so $V_1 = 0$ , and

$m\,V_2 = J \qquad \text{so that} \qquad V_2 = \frac{J}{m}$

Substituting numerical values gives

$V_2 = \frac{8.53}{2.0} = 4.27\,\text{m/s}$

Systems of Particles

For a system of $N$ particles with masses $m_i$ and velocities $\vec{V}_i$ , the total linear momentum is

(53) $\begin{equation*} \vec{p}_{\text{total}} = \sum_{i=1}^{N} m_i \, \vec{V}_i \end{equation*}$

The motion of the center of mass is governed by

(54) $\begin{equation*} M \, \vec{V}_{\text{cm}} = \vec{p}_{\text{total}} \end{equation*}$

where $M = \sum m_i$ is the total mass and $\vec{V}_{\text{cm}}$ is the velocity of the center of mass. When no external forces act on the system, the center of mass moves with constant velocity in an inertial frame.

During collisions and other short-duration interactions, the forces between interacting bodies are often very large but act over very short time intervals. In such cases, the impulse-momentum theorem provides the most direct means of analysis. When external impulses are negligible, conservation of linear momentum applies directly to the interacting bodies, regardless of the details of the internal force history during the interaction. The following example illustrates these principles for a system in which large internal impulsive forces act over a very short time interval.

Consider an A-10 aircraft firing a short burst from its nose-mounted GAU-8 cannon while in level flight. Over the very short firing interval, external forces in the flight direction (thrust and drag) can be neglected relative to the large internal forces among the gun, the aircraft structure, and the projectiles. Therefore, the aircraft plus the fired rounds may be treated as an isolated system along the flight direction.

Let the aircraft mass be $m_A$ , the mass of each projectile be $m_b$ , the number of rounds fired be $N$ , the initial aircraft speed be $V_0$ , the aircraft speed after firing be $V_A$ , and the projectile exit speed be $V_b$ . All velocities are measured along the flight direction in an inertial reference frame fixed to the Earth. Before firing, the total linear momentum of the system is

$p_{\text{total},\, i} = (m_A + N \, m_b) V_0$

Immediately after the burst, the total momentum is

$p_{\text{total},\, f} = m_A \, V_A + N \, m_b \, V_b$

With negligible external force in the flight direction during the short firing interval, conservation of linear momentum gives

$m_A \, V_A + N \, m_b \, V_b = (m_A + N \, m_b) V_0$

Solving for the aircraft speed after firing gives

$V_A = V_0 + \frac{N \, m_b}{m_A}(V_0 - V_b)$

Because $V_b \gg V_0$ , the quantity $V_0 - V_b$ is negative and $V_A < V_0$ , so the aircraft experiences a recoil-induced reduction in forward speed.

Angular Momentum & Rotational Motion

Linear momentum describes translational motion, while angular momentum describes rotational motion. Together, they provide a complete mechanical description of how particles and rigid bodies move under the action of forces and torques. Angular momentum arises naturally from Newton’s laws whenever a force acts with a moment arm relative to a reference point or axis. Its conservation principle is one of the most powerful in mechanics and underlies the dynamics of rotating machinery, rigid-body systems, and orbital motion.

The moment of a force about a reference point $O$ , also called the torque, measures the tendency of a force to produce rotational motion. If a force $\vec{F}$ acts at a point whose position vector relative to $O$ is $\vec{r}$ , the torque is defined by

(55) $\begin{equation*} \boldsymbol{\tau} = \vec{r} \times \vec{F} \end{equation*}$

The magnitude of the torque equals the product of the force and the perpendicular distance from the reference point to the line of action of the force. Only the component of the force perpendicular to $\vec{r}$ contributes to the torque, as shown in the figure below.

The creation of torques and the production of rotational motion are intrinsic to many aerospace problems.

The angular momentum of a particle about a reference point $O$ is defined as the moment of its linear momentum about that point. If the particle has linear momentum $\vec{p} = m\vec{V}$ and position vector $\vec{r}$ , then

(56) $\begin{equation*} \vec{L} = \vec{r} \times \vec{p} = m \left( \vec{r} \times \vec{V} \right) \end{equation*}$

Angular momentum is a vector that is perpendicular to the plane formed by $\vec{r}$ and $\vec{p}$ , with the orientation given by the right-hand rule. Differentiating this definition with respect to time and applying Newton’s second law in momentum form gives the rotational equation of motion for a particle, i.e.,

(57) $\begin{equation*} \sum \boldsymbol{\tau} = \frac{d\vec{L}}{dt} \end{equation*}$

This relationship is the rotational analogue of $\sum \vec{F} = d\vec{p}/dt$ for translational motion.

If the net external torque acting on a particle or a system of particles about a reference point is zero, then

(58) $\begin{equation*} \sum \boldsymbol{\tau}_{\text{ext}} = \vec{0} \end{equation*}$

and

(59) $\begin{equation*} \frac{d\vec{L}_{\text{total}}}{dt} = \vec{0} \end{equation*}$

so that the total angular momentum is constant, i.e., $\vec{L}_{\text{total}}$ = constant. This conservation law governs the dynamics of rotating bodies, interacting systems, and orbital motion.

For a rigid body rotating about a fixed axis, all its mass elements move in circular paths about that axis. If $\omega$ denotes the angular velocity and $r$ the perpendicular distance of a mass element $dm$ from the axis, the local speed is $V = \omega r$ . The angular momentum of the rigid body about the axis may be written as

(60) $\begin{equation*} \vec{L} = I\boldsymbol{\omega} \end{equation*}$

where the rotational or polar moment of inertia $I$ is defined by

(61) $\begin{equation*} I = \int r^{2}\, dm \end{equation*}$

The moment of inertia is the rotational analogue of mass and depends entirely on the distribution of mass relative to the axis of rotation.

For rigid-body rotation about a fixed axis, the rotational form of Newton’s second law becomes

(62) $\begin{equation*} \sum \tau = \frac{dL}{dt} \end{equation*}$

Using $L = I\omega$ , this reduces to

(63) $\begin{equation*} \sum \tau = I \,\frac{d\omega}{dt} \end{equation*}$

or

(64) $\begin{equation*} \sum \tau = I \, \alpha \end{equation*}$

where $\alpha = d\omega/dt$ is the angular acceleration. This latter equation plays the same fundamental role in rotational dynamics that $\sum F = m \, a$ plays in translational motion.

A rigid body in rotation possesses kinetic energy associated with its angular motion. The rotational kinetic energy is

(65) $\begin{equation*} T_{\text{rot}} = \frac{1}{2} \, I \, \omega^{2} \end{equation*}$

which is directly analogous to the translational kinetic energy $T = \frac{1}{2} \, m\, V^{2}$ . When a torque $\tau$ produces an angular displacement $d\theta$ , the differential work done by the torque is

(66) $\begin{equation*} dW = \tau\, d\theta \end{equation*}$

and the instantaneous power associated with the torque is

(67) $\begin{equation*} P = \tau\,\omega \end{equation*}$

Angular momentum and torque extend Newtonian mechanics from pure translation to rotation in a consistent manner. The relationships among torque, angular momentum, moment of inertia, angular acceleration, work, and power underlie the analysis of many engineering systems. The conservation of angular momentum, in particular, is one of the central invariance principles of classical mechanics.

Consider an airplane with a single turboshaft engine starting on the ground. The propeller is initially at rest and is accelerated to an operating angular speed $\omega_0$ about the shaft axis by the torque produced by the engine. During this start-up transient, the propeller behaves approximately as a rigid body rotating about a fixed axis. Let the propeller have a polar moment of inertia $I$ about the shaft. Assume that the engine delivers an approximately constant torque $\tau$ during the spin-up.

The rotational equation of motion is

$\sum \tau = I \, \alpha$

With a constant applied torque, then

$\alpha = \frac{\tau}{I}$

so the angular velocity satisfies

$\frac{d\omega}{dt} = \frac{\tau}{I}$

Integrating from rest ( $\omega = 0$ at $t = 0$ ) gives

$\omega(t) = \frac{\tau}{I}\, t$

Therefore, the time required to reach the operating speed $\omega_0$ is

$t_s = \frac{I \, \omega_0}{\tau}$

At the operating speed, the propeller’s angular momentum about the shaft is

$L = I \, \omega_0$

and the rotational kinetic energy stored in the spinning propeller is

$T_{\text{rot}} = \frac{1}{2} I \, \omega_0^{2}$

During the spin-up, the instantaneous mechanical power delivered by the engine to the propeller is

$P = \tau \, \omega$

Check Your Understanding #5 – Angular Momentum

A small mass $m$ = 1.5 kg moves on a frictionless horizontal table in a circular path of radius $r_1$ = 0.60 m with speed $V_1$ = 4.0 m/s. The mass is attached to a light, inextensible string that passes through a small hole in the table. The string is pulled slowly so that the radius is reduced to $r_2$ = 0.25 m. Neglect any external torque about the hole. Determine the mass’s new speed and the change in its kinetic energy.

Show solution/hide solution.

Because no external torque acts about the hole, the angular momentum of the mass about that point is conserved. The angular momentum is

$L = m\,r\,V$

so conservation gives

$m\,r_1\,V_1 = m\,r_2\,V_2$

and so

$V_2 = \frac{r_1}{r_2}\,V_1$

Substituting numerical values gives

$V_2 = \frac{0.60}{0.25}\,(4.0) = 9.6\,\text{m/s}$

The initial kinetic energy is

$T_1 = \frac{1}{2}\,m\,V_1^2$

$T_1 = \frac{1}{2}(1.5)(4.0^2) = 12.0\,\text{J}$

The final kinetic energy is

$T_2 = \frac{1}{2}\,m\,V_2^2 = \frac{1}{2}(1.5)(9.6^2) = 69.1\,\text{J}$

Therefore, the change in kinetic energy is

$\Delta T = T_2 - T_1 = 57.1\,\text{J}$

Gravitation & Central-Force Motion

Gravitation is the fundamental interaction that governs the motion of bodies on astronomical scales and plays a central role in orbital motion and many large-scale mechanical phenomena. In classical mechanics, gravitation is described as a central force acting between masses. Motion under an inverse-square force law provides one of the most complete applications of Newton’s laws, momentum, energy, and angular momentum.

Newton’s Law of Universal Gravitation

Newton’s law of universal gravitation states that every pair of point masses attracts one another with a force whose magnitude is proportional to the product of their masses and inversely proportional to the square of the distance between them. If two point masses $m_1$ and $m_2$ are separated by a distance $r$ , the magnitude of the gravitational force is

(68) $\begin{equation*} F = G \left( \frac{m_1 \, m_2}{r^2} \right) \end{equation*}$

where $G$ is the universal gravitational constant. The force acts along the line joining the two masses and is always attractive.

Gravitational Field & Weight

The gravitational field $\vec{g}$ produced by a mass $M$ at a distance $r$ is defined as the gravitational force per unit mass acting on a test particle

(69) $\begin{equation*} \vec{g} = - G \left( \frac{M}{r^2} \right) \hat{\vec{r}} \end{equation*}$

where $\hat{\vec{r}}$ is the outward radial unit vector. The gravitational force on a particle of mass $m$ placed in this field is

(70) $\begin{equation*} \vec{F} = m \, \vec{g} \end{equation*}$

In general, for two masses $m_1$ and $m_2$ separated by a distance $R$ , then

(71) $\begin{equation*} F_{\text{grav}} = - \frac{G \, m_1 \, m_2}{R^2} \end{equation*}$

where $G$ is the gravitational constant. This is called the universal law of gravitation.

The universal law of gravitation states that the force is proportional to the product of their masses and inversely proportional to the square of their separation distance.

Near the surface of the Earth, the magnitude of the gravitational field is nearly constant and is denoted by ${g_0}$ . The gravitational force is then written as

(72) $\begin{equation*} W = m \, g_0 \end{equation*}$

This force is called the body’s weight. Remember that mass is a measure of inertia, i.e., a resistance to being sped up, slowed down, or changed in direction, whereas weight is a force that depends on gravity. The familiar value of ${g_0}$ = 9.81 m/s $^2$ (or 32.17 ft/s $^2$ arises directly from Newton’s law of gravitation applied at the surface of the Earth. For a mass $m$ located on the Earth’s surface and using Newton’s second law, $F = m \, a$ , then

(73) $\begin{equation*} F_g = - \frac{G \, M_E \, m}{R_E^2} = - m \, g_0 \end{equation*}$

where $M_E$ is the mass of the Earth and $R_E$ is its mean radius. The Earth is not a perfect sphere but an oblate spheroid; however, the spherical approximation is sufficiently accurate.

The mass and radius of the Earth determine the acceleration under gravity at the surface

With $M_E$ = 5.97 $\times$ 10 $^{24}$ kg, $R_E$ = 6.3781 $\times$ 10 $^{6}$ , and $G$ = 6.67428 $\times$ 10 $^{-11}$ N m ${^2}$ kg $^{-2}$ , then Eq. 73 gives

(74) $\begin{equation*} g_0 = \frac{G \, M_E}{R_E^2} \end{equation*}$

Substituting the numerical values gives

(75) $\begin{equation*} g_0 = \frac{ (6.67428 \times 10^{-11}) (5.97 \times 10^{24}) }{ (6.3781 \times 10^6)^2 } = 9.81~\text{m/s}^2 \end{equation*}$

Therefore, the standard value of the gravitational acceleration, ${g_0}$ , at the Earth’s surface follows directly from the universal law of gravitation.

Gravitational Potential Energy

The gravitational force is conservative and so possesses a scalar potential-energy function. For two point masses $M$ and $m$ separated by a distance $r$ , the gravitational potential energy is

(76) $\begin{equation*} U(r) = - G \left( \frac{M \, m}{r} \right) \end{equation*}$

The negative sign reflects the attractive nature of the gravitational force and the choice of zero potential energy at infinite separation.

Near the surface of the Earth, where the gravitational field may be treated as uniform, the gravitational potential energy reduces to the familiar approximation

(77) $\begin{equation*} U = m \, g \, h \end{equation*}$

where $h$ is the vertical elevation measured from a chosen reference level.

Central-Force Motion

A force that always acts along the line joining a particle to a fixed point and whose magnitude depends only on the distance from that point is called a central force. Gravitation is the most important example of a central force in classical mechanics. Motion under a central force is always confined to a plane, and the angular momentum of the particle about the force center is conserved.

For a particle of mass $m$ acted upon by a central force $\vec{F}(r) = F(r) \, \hat{\vec{r}}$ , conservation of angular momentum implies

(78) $\begin{equation*} \vec{L} = \vec{r} \times m \, \vec{V} = \text{constant} \end{equation*}$

This result is independent of the particular functional form of $F(r)$ and follows from the vanishing of the torque for a central force.

Equation of Motion in a Gravitational Field

For motion under the gravitational attraction of a fixed mass $M$ , Newton’s second law gives

(79) $\begin{equation*} m \, \overbigddot{\vec{r}} = - G \left( \frac{M \, m}{r^2} \right) \hat{\vec{r}} \end{equation*}$

Dividing by $m$ yields the acceleration field

(80) $\begin{equation*} \overbigddot{\vec{r}} = - G \left( \frac{M}{r^2} \right) \hat{\vec{r}} \end{equation*}$

This second-order vector differential equation governs all motion in a gravitational field.

Because the gravitational force is conservative, the total mechanical energy of a particle moving under gravity is conserved. The total energy is the sum of kinetic and gravitational potential energies, i.e.,

(81) $\begin{equation*} E = \frac{1}{2} \, m \, V^2 - G \left( \frac{M \, m}{r} \right) \end{equation*}$

The invariance of this quantity provides an alternative to direct integration of the equations of motion and plays a central role in orbital mechanics and analysis.

Inverse-Square Law & Orbital Paths

Motion under an inverse-square central force leads to trajectories that are conic sections. The possible paths include circles, ellipses, parabolas, and hyperbolas. The specific form of the trajectory depends on the particle’s total mechanical energy and angular momentum through

(82) $\begin{equation*} r(\theta) = \frac{h^2 / \mu}{1 + e \, \cos\theta} \end{equation*}$

where $h$ is the conserved angular momentum per unit mass, $\mu = G \, M$ is the gravitational parameter, and $e$ is the orbital eccentricity. For $0 \le e < 1$ , the orbit is an ellipse with the gravitating body at a focus, as shown in the figure below.

A classic conic section is an elliptical orbit of a planet or satellite with the gravitating body at the focus.

Because the torque about the force center vanishes, angular momentum is conserved, and the areal velocity is constant, i.e.,

(83) $\begin{equation*} \frac{dA}{dt} = \frac{h}{2} \end{equation*}$

which is Kepler’s second law. Conservation of mechanical energy further gives, for elliptical motion, i.e.,

(84) $\begin{equation*} E = - \frac{\mu}{2 \, a} \end{equation*}$

and the orbital period

(85) $\begin{equation*} T = \frac{2 \, \pi \, a^{3/2}}{\sqrt{\mu}} \end{equation*}$

so that

(86) $\begin{equation*} T^2 \ \propto \ a^3 \end{equation*}$

which is Kepler’s third law. Therefore, the combination of Newton’s second law, conservation of energy, and conservation of angular momentum yields a complete and predictive description of motion in a gravitational field. It forms the physical basis for orbital mechanics and planetary dynamics.

Check Your Understanding #6 – Gravitational principles

A spacecraft of mass $m$ = 1,200 kg moves radially outward from the center of the Earth under the influence of gravity alone. At a distance $r_1$ = 7.0 $\times$ 10 $^6$ m from the Earth’s center, its speed is $V_1$ = 6.0 $\times$ 10 $^3$ m/s. Determine its speed $V_2$ when it reaches a distance $r_2$ = 1.40 $\times$ 10 $^7$ m. Use $G \, M$ = 3.986 $\times$ 10 $^{14}$ m $^3$ /s $^2$ .

Show solution/hide solution.

Because gravity is a conservative central force, the spacecraft’s total mechanical energy is conserved. The total energy at any radius is

$E = \frac{1}{2}\,m\,V^2 - \frac{G\,M\,m}{r}$

Energy conservation between $r_1$ and $r_2$ gives

$\frac{1}{2}\,m\,V_1^2 - \frac{G\,M\,m}{r_1} = \frac{1}{2}\,m\,V_2^2 - \frac{G\,M\,m}{r_2}$

The mass cancels, giving

$\frac{1}{2}\,V_1^2 - \frac{G\,M}{r_1} = \frac{1}{2}\,V_2^2 - \frac{G\,M}{r_2}$

Solving for $V_2^2$ ,

$V_2^2 = V_1^2 + 2\,G\,M \left( \frac{1}{r_2} - \frac{1}{r_1} \right)$

Substituting numerical values gives

$V_2^2 = (6.0 \times 10^3)^2 + 2(3.986 \times 10^{14}) \left( \frac{1}{1.40 \times 10^7} - \frac{1}{7.0 \times 10^6} \right) = 2.09 \times 10^7$

so $V_2$ = 4.57 $\times 10^3$ m/s.

Electricity & Magnetism

Electricity and magnetism describe the behavior of electric charge and its interactions through electric and magnetic fields and together form the unified field of electromagnetism. Aerospace engineers rely on electromagnetic principles because electrical power generation and distribution, avionics, navigation, communication, sensing, actuation, and propulsion all depend directly on an understanding of electric and magnetic fields.

Electric Charge & Coulomb’s Law

Electric charge is a fundamental property of matter that determines how bodies interact through the electromagnetic force. Unlike mass, which is always positive, electric charge exists in two distinct forms, conventionally labeled positive and negative. Like charges repel, whereas unlike charges attract. These interactions are long-range and act over space without physical contact, in direct analogy to gravitational forces.

The quantitative law governing the electrostatic force between two stationary point charges was established by Coulomb. For two charges $q_1$ and $q_2$ separated by a distance $r$ , the magnitude of the force is

(87) $\begin{equation*} F = k \left( \frac{|q_1 \, q_2|}{r^2} \right) \end{equation*}$

where $k$ is Coulomb’s constant. The force acts along the line joining the charges and varies inversely with the square of the distance. The force between like charges is repulsive, whereas the force between unlike charges is attractive.

Coulomb’s law is mathematically analogous to Newton’s law of universal gravitation, but it describes interactions that are typically many orders of magnitude stronger and that admit both attraction and repulsion. It serves as the foundational experimental law of electrostatics and provides the starting point for defining the electric field and electrical potential, as well as the entire framework of classical electromagnetism.

Electric & Magnetic Fields

Electric and magnetic fields provide a unified, field-based description of electromagnetic interactions. A stationary electric charge produces an electric field that acts on other charges, whether they are at rest or in motion. Moving charges and electric currents produce magnetic fields that act only on moving charges. The force experienced by a charged particle depends on which fields are present and on the particle’s velocity, as illustrated in the figure below.

Electric fields change particle energy, while magnetic fields change particle direction. The total force is the vector sum of the forces.

If a charge $q$ is placed in an electric field $\vec{E}$ in the absence of any magnetic field, the force acting on the charge is

(88) $\begin{equation*} \vec{F}_E = q \, \vec{E} \end{equation*}$

This electric force acts parallel to the field and is independent of the particle’s velocity. As a result, electric fields can change both the magnitude and direction of the particle’s velocity, perform work on charges, and alter their kinetic energy.

If no electric field is present, but the charge moves with velocity $\vec{V}$ through a magnetic field $\vec{B}$ , the force on the charge is

(89) $\begin{equation*} \vec{F}_B = q \, \vec{V} \times \vec{B} \end{equation*}$

The magnetic force is perpendicular to both the velocity and the magnetic field. Because it is always perpendicular to the velocity, it does no work on the charge and does not change the particle’s speed. Instead, it alters only the direction of motion, producing curved trajectories such as circular or helical paths.

In the general case, when both electric and magnetic fields are present, the total force acting on a moving charge is the vector sum of the electric and magnetic contributions. This combined interaction is described by the Lorentz force law, i.e.,

(90) $\begin{equation*} \vec{F} = q \left( \vec{E} + \vec{V} \times \vec{B} \right) \end{equation*}$

This equation shows explicitly that electric and magnetic fields influence charged particles in fundamentally different ways. Electric fields govern changes in kinetic energy, whereas magnetic fields govern the geometry of particle motion. Together, they determine the full dynamics of charged particles in electromagnetic systems.

Electric Potential, Energy, & Current

For stationary charge distributions, the electric field is conservative and can be expressed in terms of a scalar potential function $\phi$ , i.e.,

(91) $\begin{equation*} \vec{E} = - \nabla \phi \end{equation*}$

The electric potential provides an energy-based description of electrostatic interactions, analogous to gravitational potential in mechanics. It represents the electric potential energy per unit charge at each point in space.

A charge $q$ placed at a location where the electric potential is $\phi$ possesses electric potential energy, i.e.,

(92) $\begin{equation*} V = q \, \phi \end{equation*}$

Only differences in electric potential have physical significance because only changes in potential energy can influence the motion of charges. Spatial variations of electric potential give rise to electric fields through the $\vec{E} = - \nabla \phi$ term, which exert forces on charges and perform work on them. In electrical systems, these potential differences are called voltage.

When an electric potential difference is applied across a conducting material, charges are driven into motion by the resulting electric field. The ordered motion of electric charge constitutes an electric current. If a net charge $Q$ passes through a given cross-section in a time interval $dt$ , the current is defined as

(93) $\begin{equation*} I = \frac{dQ}{dt} \end{equation*}$

Therefore, electric current represents the dynamical response of charges to electric fields established by potential differences.

Electric charge is conserved in all physical processes. Charge cannot be created or destroyed; it can only be transferred from one body or region to another. This fundamental conservation law underlies both circuit and electromagnetic field theories and ensures that currents entering and leaving any region are correctly balanced.

Electromagnetic Induction

Electric and magnetic fields are dynamically coupled. In particular, a time-varying magnetic field generates an electric field, even in regions of space containing no electric charge. This phenomenon, known as electromagnetic induction, represents a fundamental departure from electrostatics, in which electric fields arise only from charge distributions.

The quantitative statement of electromagnetic induction is given by Faraday’s law. In integral form, it states that the circulation of the electric field around any closed curve equals the negative time rate of change of the magnetic flux through any surface bounded by that curve, i.e.,

(94) $\begin{equation*} \oint \vec{E} \bigcdot d\vec{s} = - \frac{d}{dt} \int_S \vec{B} \bigcdot d\vec{S} \end{equation*}$

The surface integral defines the magnetic flux through the surface, i.e.,

(95) $\begin{equation*} \Phi_B = \int_S \vec{B} \bigcdot d\vec{S} \end{equation*}$

Therefore, a changing magnetic flux produces an electric field whose field lines form closed loops. Unlike electrostatic fields, induced electric fields are not conservative and do not originate or terminate at charges. Consequently, an electric potential cannot be uniquely defined in regions where electromagnetic induction is present.

Electromagnetic induction can arise in two physically distinct but mathematically equivalent ways. In the first, the magnetic field varies with time while the conducting circuit remains stationary. In the second, known as the motional electromotive force (emf), a conductor moves through a magnetic field. The basic mechanism of motional emf may be illustrated by a conducting rod moving with velocity ${\vec{V} }$ through a uniform magnetic field $\vec{B}$ that is perpendicular to both the rod and its direction of motion, as shown in the figure below. The mobile charge carriers within the conductor experience the magnetic component of the Lorentz force given by

(96) $\begin{equation*} \vec{F}_B = q \, \vec{V} \times \vec{B} \end{equation*}$

Motion through a magnetic field, or through a time-varying magnetic flux, induces an electric field and an emf.

This force drives charge carriers along the length of the rod, leading to a separation of charge between its ends. As charge accumulates, an electric field is established inside the conductor. In the steady state, the electric force balances the magnetic force, i.e.,

(97) $\begin{equation*} q \vec{E} = q \, \vec{V} \times \vec{B} \end{equation*}$

so that no further net charge motion occurs along the rod. The resulting electric field gives rise to a measurable potential difference between the ends of the conductor, which is the motional emf.

The negative sign in Faraday’s law expresses Lenz’s law. It indicates that the induced electric field acts in a direction that opposes the change in magnetic flux that produces it. This opposition is a direct consequence of energy conservation and prevents spontaneous amplification of electromagnetic energy.

Faraday’s law is a local field law and applies equally in free space and within conducting materials. In differential form, it is written as

(98) $\begin{equation*} \nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t} \end{equation*}$

This equation shows that the curl of the electric field equals the negative time derivative of the magnetic field. Together with the other Maxwell equations, it establishes that electric and magnetic fields are inseparably linked and can propagate through space as electromagnetic waves. Electromagnetic induction provides the physical basis for electrical generators, motors, transformers, inductors, magnetos, and all forms of electromagnetic wave propagation.

Maxwell’s Equations & Electromagnetic Fields

Maxwell’s equations provide the complete classical description of electric and magnetic fields. These four coupled partial differential equations relate the electric field $\vec{E}(\vec{x},t)$ and magnetic field $\vec{B} (\vec{x},t)$ to their physical sources, i.e., electric charge density $\rho(\vec{x},t)$ and electric current density $\vec{J}(\vec{x},t)$ . Together, they describe how electric and magnetic fields are generated, how they evolve in time, and how they interact with matter.

In differential form, Maxwell’s equations in vacuum are

(99) $\begin{equation*} \nabla \bigcdot \vec{E} = \frac{\rho}{\varepsilon_0}, \qquad \nabla \bigcdot \vec{B} = 0, \qquad \nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t}, \qquad \nabla \times \vec{B} = \mu_0 \, \vec{J} + \mu_0 \, \varepsilon_0 \frac{\partial \vec{E}}{\partial t} \end{equation*}$

The first equation (Gauss’s law) states that electric charge is the source of the electric field. The second equation expresses the absence of magnetic monopoles, implying that magnetic field lines form closed loops. The third equation (Faraday’s law) shows that a time-varying magnetic field necessarily produces a circulating electric field. The fourth equation (Ampère-Maxwell law) states that magnetic fields are generated both by electric currents and by time-varying electric fields, the latter contribution being Maxwell’s displacement current.

The constant $\mu_0$ is the magnetic permeability of free space (vacuum). It defines the proportionality between the magnetic flux density $\vec{B}$ and the magnetic field intensity $\vec{H}$ in vacuum according to

(100) $\begin{equation*} \vec{B} = \mu_0 \, \vec{H} \end{equation*}$

The SI units of $\mu_0$ are henries per meter (H/m), equivalent to newtons per ampere squared (N/A $^2$ ). Its numerical value is $\mu_0 = 4\pi \times 10^{-7} \ \text{H/m}$ . The electric permittivity of free space $\varepsilon_0$ plays an analogous role for electric fields, relating the electric displacement field $\vec{D}$ to the electric field $\vec{E}$ in vacuum through the constitutive relation $\vec{D} = \varepsilon_0 \, \vec{E}$ . Together, $\mu_0$ and $\varepsilon_0$ define the intrinsic electromagnetic properties of free space.

A fundamental consequence of Maxwell’s equations is that coupled electric and magnetic field disturbances can propagate through space even in the absence of charges and currents. Combining the curl equations leads to wave equations for $\vec{E}$ and $\vec{B}$ with propagation speed

(101) $\begin{equation*} c = \frac{1}{\sqrt{\mu_0 \, \varepsilon_0}} \end{equation*}$

This result identifies light as an electromagnetic wave. For a plane electromagnetic wave propagating in a vacuum, the electric and magnetic field magnitudes satisfy

(102) $\begin{equation*} \frac{E}{B} = c \end{equation*}$

with $\vec{E}$ and $\vec{B}$ mutually perpendicular and perpendicular to the direction of propagation.

Maxwell’s equations unify electrostatics, magnetostatics, electromagnetic induction, and current-generated magnetic fields into a single self-consistent field theory. When combined with the Lorentz force law, i.e.,

(103) $\begin{equation*} \vec{F} = q \left( \vec{E} + \vec{V} \times \vec{B} \right) \end{equation*}$

they provide a complete classical description of how electromagnetic fields interact with charged matter. In aerospace engineering, Maxwell’s equations govern the behavior of antennas, radar systems, wireless communication links, electromagnetic sensors, power-distribution systems, and spacecraft interactions with ionized plasmas and space environments.

Energy in Electromagnetic Fields

Electric and magnetic fields are not merely intermediaries that transmit forces between charges and currents; they possess energy in their own right and can transport that energy through space. The energy density stored in an electromagnetic field is given by

(104) $\begin{equation*} u = u_E + u_B = \frac{1}{2} \, \varepsilon_0 \, E^2 + \frac{1}{2 \, \mu_0} B^2 \end{equation*}$

The first term represents energy stored in the electric field, while the second represents energy stored in the magnetic field. This expression shows that electromagnetic energy is a local field quantity, defined at every point in space and time, and does not require the presence of matter. A static electromagnetic field is one in which the electric and magnetic field components are time-independent. In such fields, electromagnetic energy is stored but is not transported through space. When the fields vary with time, electromagnetic energy can be transported through space as radiation.

The Poynting vector describes the flow of electromagnetic energy through space and is defined as the electromagnetic energy flux, that is, the rate of energy transport per unit area, i.e.,

(105) $\begin{equation*} \vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B} \end{equation*}$

The magnitude of the time-averaged Poynting vector gives the energy flux carried by electromagnetic radiation. This quantity plays a central role in relating electromagnetic energy transport to momentum transfer and radiation pressure. A pulse of radiation with energy $E$ carries a momentum of

(106) $\begin{equation*} p = \frac{E}{c} \end{equation*}$

in the direction of propagation. This relation is required for momentum conservation and is experimentally confirmed by radiation pressure. For normally incident radiation absorbed by a surface, the radiation pressure is

(107) $\begin{equation*} P = \frac{I}{c} \end{equation*}$

where $I$ is the energy flux. Therefore, the Poynting vector gives both the magnitude and the direction of electromagnetic power transport, representing the power flow per unit area carried by the fields, as shown in the figure below. Its divergence determines the local rate of conversion between electromagnetic field energy and mechanical or thermal energy in matter, which is the content of Poynting’s theorem.

An electromagnetic wave traveling through space.

These field-energy relations provide the physical foundation for power transmission, waveguides, antennas, microwave systems, and electromagnetic radiation. Together with the laws of mechanics and gravitation, the laws of electromagnetism complete the classical framework required for many types of engineering analysis.

Check Your Understanding #7 – Electricity

An electron is released from rest in a uniform electric field of magnitude $E$ = 4.0 $\times$ 10 $^4$ V/m. The field extends over a distance of 2.5 cm. Determine: (a) the work done by the electric field on the electron and (b) the speed of the electron after it has moved through this distance. Use $e$ = 1.60 $\times$ 10 $^{-19}$ C and $m_e$ = 9.11 $\times$ 10 $^{-31}$ kg.

Show solution/hide solution.

The electric force acting on the electron has a magnitude

$F = e\,E$

The work done by a constant force acting over distance $s$ is

$W = F\,s = e\,E\,s$

Substituting numerical values gives

$W = (1.60 \times 10^{-19})(4.0 \times 10^4)(2.5 \times 10^{-2}) = 1.60 \times 10^{-16}\,\text{J}$

The electron starts from rest, so the work done by the electric field equals the change in kinetic energy, i.e.,

$W = \frac{1}{2}\,m_e\,V^2$

Solving for the speed gives

$V = \sqrt{\frac{2\,W}{m_e}} = \sqrt{\frac{2(1.60 \times 10^{-16})}{9.11 \times 10^{-31}}} = 1.87 \times 10^7\,\text{m/s}$

Optics

In aerospace engineering, light and other electromagnetic radiation are frequently encountered under conditions in which they propagate away from their sources and interact with matter over distances that are large compared with their wavelength. In such cases, solving Maxwell’s equations is unnecessary. Instead, simpler optical descriptions can be employed that retain the underlying conservation laws while exploiting scale separation between wavelength, geometry, and material variation. These descriptions are limiting forms of electromagnetic theory appropriate when radiation transport governs the behavior of a given system. Depending on the relative scales involved, radiation may be modeled either as rays that describe the transport of energy and momentum along well-defined paths or as waves when diffraction and interference are nonnegligible. The appropriate description is determined by the wavelength relative to the geometric and material length scales.

Geometrical Optics & Ray Propagation

Geometrical optics is the asymptotic limit of electromagnetic propagation in which the wavelength is small compared with all characteristic dimensions of the system. In this limit, the electromagnetic field may be expressed locally as a rapidly varying phase multiplied by a slowly varying amplitude, and energy transport becomes confined to narrow paths whose centerlines define optical rays. The field description is thereby reduced to a kinematic description of energy propagation, analogous to particle trajectories in mechanics. These relations govern the redirection of light by lenses and mirrors, and define the mapping between object space and image space in optical systems.

Ray paths of light and other electromagnetic radiation are determined by Fermat’s principle, which states that the physical ray connecting two points is the path for which the travel time is stationary with respect to nearby paths. For radiation propagating through a medium of refractive index $n$ , this condition can be expressed in terms of the optical path length, i.e.,

(108) $\begin{equation*} \mathcal{L} = \int n \, ds \end{equation*}$

where ${ds}$ is a differential element of geometric path length along the ray. The optical path length is the distance a ray of light would travel in a vacuum in the same time required to traverse the actual path through the medium. Equivalently, it measures the total phase accumulated along the path because the phase of a monochromatic wave is proportional to $\mathcal{L}$ .

In a homogeneous medium, electromagnetic rays propagate along straight lines. Spatial variations in refractive index introduce curvature into the ray paths, causing the rays to bend toward regions of higher refractive index. In this sense, refraction and reflection at boundaries, as well as gradual bending in continuously varying media, follow from the same underlying requirement that the physical ray is selected by the optical travel time (or optical phase accumulation).

Consider two homogeneous media that are separated by a planar interface, with a fixed source point $A$ and observation point $B$ , as shown in the figure below. A typical ray consists of two straight segments, $A \rightarrow P$ in medium 1 with refractive index ${n_1}$ and $P \rightarrow B$ in medium 2 with refractive index $n_2$ . For a given location of $P$ , the optical path length is

(109) $\begin{equation*} \mathcal{L}(P) = n_1 \, AP + n_2 \, PB = n_1 \, l_1 + n_2 \, l_2 \end{equation*}$

where $AP$ and $PB$ denote the geometric lengths of the two segments, $l_1$ and $l_2$ , respectively. Fermat’s principle requires that the physical ray correspond to the location of $P$ for which $\mathcal{L}(P)$ is stationary with respect to small displacements of $P$ along the interface. Enforcing this condition yields the law of refraction, or Snell’s law, i.e.,

(110) $\begin{equation*} n_1 \sin \theta_1 = n_2 \sin \theta_2 \end{equation*}$

where $\theta_1$ and $\theta_2$ are the angles that the incident and refracted rays make with the normal to the interface.

Geometry of a light ray used to obtain Snell’s law from Fermat’s principle.

This latter result can also be obtained from the continuity of phase along the interface. For a monochromatic wave of angular frequency $\omega$ , the wave number magnitude in medium $i$ is

(111) $\begin{equation*} k_i = \frac{n_i \, \omega}{c} \end{equation*}$

where $c$ is the speed of light in vacuum. Along the interface, the phase variation must be the same in both media, which requires equality of the tangential components of the wave vectors, i.e.,

(112) $\begin{equation*} k_1 \sin \theta_1 = k_2 \sin \theta_2 \end{equation*}$

Substituting $k_i = n_i \omega / c$ and cancelling the common factor $\omega/c$ then leads to Snell’s law.

Reflection occurs when radiation remains in the same medium, and no transmitted wave is supported in the adjacent region. In this case, the incident and reflected waves have the same magnitude. Continuity of phase along the interface requires equality of the tangential components of the incident and reflected wave vectors, i.e.,

(113) $\begin{equation*} k \sin \theta_i = k \sin \theta_r \end{equation*}$

so that $\theta_i = \theta_r$ , where $\theta_i$ and $\theta_r$ are the angles of incidence and reflection measured from the interface normal. Therefore, the law of reflection follows from the same phase-matching requirement as refraction.

Lenses and mirrors are optical elements that redirect radiation by imposing controlled changes in the direction of rays at material interfaces. A lens consists of one or more refracting surfaces whose curvature causes rays to converge or diverge according to Snell’s law, while a mirror redirects rays by reflection at a shaped surface. A convex lens, which is thicker at its center than at its edges, causes initially parallel rays to converge toward a focal point, whereas a concave lens causes rays to diverge.

The foregoing characteristics follow directly from the local refraction of rays at each surface and may be analyzed quantitatively using geometrical optics. By combining multiple refracting and reflecting surfaces, optical systems may be designed to form images, collimate radiation, or control beam paths.

Radiative Transfer & Atmospheric Optics

When electromagnetic radiation propagates through gases, plasmas, or particle-laden flows, its interaction with matter cannot always be described by ray geometry alone. In such environments, radiation exchanges energy with the medium through absorption, emission, and scattering, so energy transport must be described using conservation principles rather than purely geometric ray paths.

The relevant quantity is the radiative intensity, which represents the directional flow of radiant energy and evolves according to a balance between propagation and interaction with the surrounding medium. At a given point in space, the spectral radiative intensity $I_\nu$ is defined such that the differential radiant power $dP$ crossing an area element $dA$ , oriented normal to the propagation direction, within a differential solid angle $d\Omega$ about that direction and within a differential frequency band $d\nu$ , is

(114) $\begin{equation*} dP = I_\nu \, dA \, d\Omega \, d\nu \end{equation*}$

Therefore, radiative intensity quantifies the energy carried by radiation in a specified direction, rather than the local amplitude of the electromagnetic field. In the absence of interaction with matter, $I_\nu$ remains constant along a ray. Absorption, emission, and scattering occur only when radiation interacts with a medium, at which point intensity serves as a convenient bookkeeping variable for enforcing conservation of radiant energy.

Because absorption, emission, and scattering depend on temperature, density, composition, and particulate content, radiation transport in aerospace applications is often coupled directly to fluid mechanics and thermodynamics. In regimes where these interactions are significant, such as atmospheric sensing, high-altitude flight, and high-speed flows, ray optics alone is insufficient and must be supplemented by radiative energy balance models.

Check Your Understanding #8 – Geometrical optics in PIV measurements

A collimated green laser beam used to form a PIV light sheet in air ${n_1}$ = 1.00 is incident on a flat glass window. The refractive index of glass varies weakly with wavelength (dispersion); for this calculation, use the value appropriate for green light, $n_2$ = 1.52. The beam makes an angle of 30 $^\circ$ with the surface normal. (a) Determine the angle between the refracted beam and the normal to the glass surface inside the glass. (b) State whether the refracted ray bends toward or away from the normal and explain why. (c) If the glass window has parallel faces, describe the direction of the emerging beam relative to the incident beam after it exits the second surface.

Show solution/hide solution.

(a) The use of Snell’s law gives

$n_1 \sin \theta_1 = n_2 \sin \theta_2$

so in this case

$\sin \theta_2 = \frac{n_1}{n_2}\sin \theta_1 = \frac{1.00}{1.52}\sin 30^\circ = \frac{1.00}{1.52}(0.5) = 0.329$

therefore

$\theta_2 = \sin^{-1}(0.329) \approx 19.2^\circ$

(b) The ray bends toward the normal because $n_2 > n_1$ , so Snell’s law requires $\sin\theta_2 < \sin\theta_1$ , which implies $\theta_2 < \theta_1$ .

(c) For a plane-parallel window, the beam emerging from the second surface is parallel to the incident beam, so its exit angle in air equals the entry angle, $\theta_{\text{exit}} = \theta_1$ , although the beam is also laterally displaced.

Relativity Physics

Classical Newtonian mechanics assumes absolute space and absolute time and treats mass, energy, and momentum as distinct quantities. These assumptions are entirely adequate for most engineering applications, including atmospheric flight and orbital mechanics, but they are not exact. Einstein’s theory of relativity^[4] showed that space and time cannot be treated as independent, absolute quantities when physical interactions propagate at a finite speed. Relativity defines the conditions under which classical mechanics is no longer strictly valid and establishes a broader framework in which Newtonian physics appears as a limiting case.

Relativity is often introduced to engineers in physics courses by emphasizing supposed inadequacies of Newtonian mechanics and invoking paradoxes and heavy mathematical formalism, even though its essential content is simply a systematic modification of the classical transformation laws. From an engineering standpoint, its role is narrow and practical: relativistic effects become important only when classical models fail to conserve energy, momentum, or time in a self-consistent manner, most notably in high-speed motion and precision timing.

In this sense, relativity enters mechanics much as compressibility enters fluid dynamics, becoming relevant only when characteristic velocities approach the speed of information propagation. In fluids, this speed is the speed of sound $a$ , which depends on the properties of the medium; in relativity, it is the speed of light $c$ , a universal invariant in vacuum. In both cases, the governing dimensionless parameter compares a characteristic velocity to the relevant signal speed, namely the Mach number $M = U/a$ in fluid mechanics and the velocity ratio $\beta = u/c$ in relativity.

Such relativistic conditions lie far outside conventional aerospace flight regimes. Nevertheless, relativistic corrections become essential in modern space systems and navigation technologies. Accordingly, the discussion of relativity in this chapter is intended primarily to clarify the limits of validity of classical mechanics in aerospace engineering. Many engineers view relativity as a form of finite-propagation-time mechanics, in which interactions are no longer assumed to occur instantaneously. Indeed, much of the perceived conceptual difficulty in understanding the physics of relativity arises not from the introduction of new physical principles, but from the recognition that one must now enforce the finite propagation of physical interactions that classical Newtonian theory treats as occurring instantaneously.

Special Relativity

Special relativity applies to inertial reference frames and governs the behavior of matter and electromagnetic radiation at high relative speeds. Its purpose is to enforce consistent conservation of energy and momentum when physical interactions propagate at a finite, invariant speed. The theory is best understood as the coordinate transformation that preserves the form of the physical laws in all inertial frames under this constraint. Its conceptual impact is often overstated; its practical importance lies in enforcing consistency rather than redefining physical reality. For example, electromagnetic radiation travels at the speed of light and carries both energy and momentum. A pulse of radiation with energy $E$ carries a momentum $p$ of

(115) $\begin{equation*} p = \frac{E}{c} \end{equation*}$

in the direction of propagation, where $c$ is the speed of light in vacuum. This relationship is required for momentum conservation and is experimentally confirmed by the existence of radiation pressure, as previously discussed.

The consequences of this result become clear when the same radiation process is analyzed in different inertial reference frames, a central feature of Einstein’s theories. Following Einstein, consider a body at rest in an inertial frame $S$ , as shown in the figure below. The body is assumed to emit two identical electromagnetic radiation pulses in opposite directions along a common axis. Let the total radiated energy be $L$ , so each pulse carries energy $L/2$ . Because the emission is symmetric, the net radiated momentum is zero, and the body experiences no recoil reaction. In frame $S$ , the body remains at rest and so loses energy $L$ .

In frame $S$ , symmetric emission of two equal radiation pulses produces zero net momentum. In a frame $S'$ moving at speed $u$ , the pulse energies differ as a consequence of the Lorentz transformation (relativistic Doppler effect).

Now consider the same emission process from a second inertial reference frame $S'$ moving at constant speed $u$ relative to $S$ along the emission axis. Define the dimensionless velocity to the speed of light as the ratio $\beta = \dfrac{u}{c}$ and $\gamma$ as

(116) $\begin{equation*} \gamma = \frac{1}{\sqrt{1-\beta^2}} \end{equation*}$

which is known as the Lorentz factor.^[5] Aerospace engineers will recognize that the Lorentz factor is the exact, fundamental version of what the Prandtl-Glauert factor later became in aerodynamics, i.e., a correction required when finite propagation speed, in this case sound speed, can no longer be ignored. The coordinates $(x,t)$ in frame $S$ and $(x',t')$ in frame $S'$ are now related by the Lorentz transformations, i.e.,

(117) $\begin{equation*} x' = \gamma (x - u \, t) \qquad \text{and} \qquad t' = \gamma \left(t - \frac{u \, x}{c^2}\right) \end{equation*}$

Time dilation follows directly from the Lorentz transformation of the time coordinate and reflects how time coordinates transform between inertial frames, i.e., time intervals measured by a moving clock are shorter than the corresponding coordinate time intervals measured in another inertial reference frame.

Associated with these coordinate transformations are corresponding transformations of energy and momentum. For a particle or radiation pulse with energy $E$ and momentum $p$ along the direction of relative motion, the transformations are

(118) $\begin{equation*} E' = \gamma (E - u \, p) \qquad \text{and} \qquad p' = \gamma \!\left(p - \frac{u \, E}{c^2}\right) \end{equation*}$

Because electromagnetic radiation satisfies $E = p \, c$ , these relations lead directly to different energies for the forward- and backward-directed radiation pulses in frame $S'$ . This energy shift is equivalent to the relativistic Doppler effect ^[6] for electromagnetic radiation.

Because light must travel at the same speed in every inertial frame, observers in relative motion can only disagree about the energy and the momentum carried by the radiation, not its speed. Therefore, the energies measured in $S'$ are

(119) $\begin{equation*} E'_{\pm} = \frac{L}{2}\,\gamma (1 \pm \beta) \end{equation*}$

where the plus sign corresponds to the pulse propagating in the direction of the relative motion and the minus sign to the pulse propagating opposite to it. Summing the two pulse energies gives the total radiated energy measured in the moving frame, i.e.,

(120) $\begin{equation*} L' = E'_+ + E'_- = \gamma \, L \end{equation*}$

Therefore, although the body emits energy $L$ in its rest frame, the total radiated energy measured in the moving frame is larger by the factor $\gamma$ . In this regard, electromagnetic radiation carries momentum equal to its energy divided by $c$ . In the moving frame, the two pulses carry unequal momenta, i.e.,

(121) $\begin{equation*} p'_{\pm} = \frac{E'_{\pm}}{c} \end{equation*}$

and momentum conservation must still hold.

Notice that when an energy source moves toward a stationary observer, the observer will measure a higher energy flux because the same emitted energy is received over a shorter time interval, i.e., which is the relativistic Doppler effect. This distinction between total energy and energy per unit time is essential. Indeed, this is the key point of relativity: the conservation of energy and momentum must hold independently in every inertial reference frame. The Lorentz transformation, therefore, requires that the energy and momentum of radiation change between frames in a coupled and self-consistent way, rather than independently.

What happens near the speed of light?

As an object approaches the speed of light, relativistic effects predicted by special relativity become significant. In a popular presentation, Carl Sagan used the image of a Vespa scooter to illustrate how radiation observed from a rapidly moving source changes with speed. As the relative velocity increases, light emitted in the forward direction is observed at progressively shorter wavelengths, while light emitted rearward is shifted to longer wavelengths. This effect is a consequence of the relativistic Doppler shift and reflects changes in the radiation’s observed energy and momentum, not a transformation of the object itself.

In the reference frame of the scooter, its physical dimensions, internal processes, and emitted radiation remain unchanged. In an inertial frame relative to which the scooter is moving, coordinate time intervals appear dilated and lengths measured along the direction of motion appear contracted, in accordance with the Lorentz transformation. The increasing energy required to accelerate the scooter further reflects the growing contribution of kinetic energy to the total energy-momentum balance, rather than an increase in the mass of the physical substance. Because the speed of light is invariant, no finite amount of energy can accelerate a material object to $c$ .

At extreme relative speeds, forward-emitted radiation may be observed in the ultraviolet or X-ray region of the spectrum, whereas rearward radiation is redshifted to longer wavelengths. These effects arise entirely from relative motion and finite signal propagation and do not imply that the object becomes radiation. Long before such speeds could be approached, practical limitations, such as structural failure or radiation damage, would occur. Therefore, the Vespa analogy is best understood as a visualization of relativistic kinematics rather than as a literal physical transformation. But it is fun to imagine what happens!

Up to this point, the discussion of relativity has concerned only the energy and momentum carried by radiation. The implications for the emitting body itself can now be determined. Let the kinetic energy of the emitting body measured in the moving frame $S'$ be $K_1$ before emission and $K_2$ after emission. Conservation of energy in frame $S'$ requires that the decrease in the body’s kinetic energy equals the excess radiated energy measured in that frame, i.e.,

(122) $\begin{equation*} K_1 - K_2 = L' - L = L(\gamma - 1) \end{equation*}$

In the low-speed limit as $\beta \ll 1$ , the Lorentz factor $\gamma = \dfrac{1}{\sqrt{1 -\beta^2}}$ may be expanded using a binomial series for small $\beta^2$ , giving

(123) $\begin{equation*} \gamma \approx 1 + \frac{1}{2}\, \beta^2 \end{equation*}$

It then follows that

(124) $\begin{equation*} K_1 - K_2 \approx \frac{1}{2} \, L \, \beta^2 = \frac{L}{2 c^2}\, u^2 \end{equation*}$

In the same low-speed limit, a small change in a body’s inertia, represented by a quantity with the dimensions of mass, produces a corresponding change in kinetic energy of the form

(125) $\begin{equation*} \Delta K = \frac{1}{2}\, \Delta m \, u^2 \end{equation*}$

Equating the two expressions in Eqs. 122 and 125 show that the loss of energy $L$ is associated with a decrease in this quantity, given by

(126) $\begin{equation*} \Delta m = \frac{L}{c^2} \end{equation*}$

Therefore, enforcing consistent conservation of energy and momentum between inertial reference frames requires that a system which loses energy $L$ must also lose a corresponding amount of inertia equal to $L/c^2$ . This energy loss must occur through energy transport out of the system, not through a process that acts directly on the mass itself.

In special relativity, this quantity of inertia is identified with the invariant (rest) mass of the system. The rest mass, $m_0$ , is defined in terms of the total energy of the system measured in its rest frame by

(127) $\begin{equation*} m_0 \equiv \frac{E_0}{c^2} \end{equation*}$

Notice that this definition is an energy relationship, not a physical conversion law. When energy is transported away from a system, for example, by electromagnetic radiation, the quantity defined as its rest mass changes accordingly. This change reflects energy bookkeeping required for consistency between the inertial reference frames, not a transformation of mass into other forms of energy.

For material bodies, this same consistency requirement leads to a coupled description of energy and momentum rather than treating them as independent quantities, as in Newtonian mechanics. In modern notation, the coupling is summarized compactly by the invariant energy-momentum relation, which means it has the same value for all inertial observers, i.e.,

(128) $\begin{equation*} E^2 = (p\,c)^2 + (m_0\,c^2)^2 \end{equation*}$

where $m_0$ is the invariant rest mass. Einstein did not present his 1905 results in this compact form, but the invariant energy-momentum relation provides a concise summary of the energy-momentum consistency that is implicit in his original radiation-based arguments.

Notice that for electromagnetic radiation, the rest mass is zero, i.e., $m_0 = 0$ , and the energy-momentum relation reduces to

(129) $\begin{equation*} E = p\,c \end{equation*}$

which recovers the radiation momentum relation used in energy-momentum balance arguments. For a material body at rest in an inertial frame, the momentum is zero, $p = 0$ , and the energy-momentum relation reduces to the classic, perhaps ubiquitous result that

(130) $\begin{equation*} E_0 = m_0 \, c^2 \end{equation*}$

This expression conveys the body’s rest energy or, more precisely, its inertial energy. It states that inertia corresponds to an energy content even when the body is at rest. But, again, this relation does not describe a physical mechanism by which mass is transformed into other forms of energy, nor does it imply that mass itself moves at the speed of light. Instead, the speed of light, $c$ , establishes the energy bookkeeping required for consistency between the inertial reference frames.

Another way of thinking about rest energy is that it is not energy “doing something,” but the energy associated with being something. Rest mass must, therefore, be viewed as a property, not a substance. By analogy, a metal spring under tension possesses more energy than the same spring when unstressed, but no mass from the metal has been converted into energy. Likewise, changes in rest mass reflect changes in a system’s energy content, not the transformation of mass into energy as a material substance. In processes such as nuclear reactions, where measurable changes in rest mass occur, the same relation quantifies the associated energy difference; however, the underlying physical mechanisms belong to nuclear physics rather than to relativity itself.

For a material body moving at speed $u$ , the invariant relationship determines the functional dependence of energy and momentum on velocity. It is convenient to write the resulting expressions as

(131) $\begin{equation*} E = \gamma \, m_0 \, c^2 \qquad \text{and} \qquad p = \gamma \, m_0 \, u \end{equation*}$

where

(132) $\begin{equation*} \gamma = \frac{1}{\sqrt{1 - u^2/c^2}} = \frac{1}{\sqrt{1 - \beta^2}} \end{equation*}$

Substitution of these expressions into the invariant relation verifies that it is satisfied identically for all inertial observers. In the low-speed limit as $\beta \ll 1$ , the relativistic energy expression becomes

(133) $\begin{equation*} E \approx m_0 \, c^2 \left( 1 + \frac{1}{2}\, \beta^2 \right) = m_0 \, c^2 + \frac{1}{2} \, m_0 \, c^2 \, \beta^2 \end{equation*}$

and using $\beta = u/c$ , this result becomes

(134) $\begin{equation*} E \approx m_0 \, c^2 + \frac{1}{2} \, m_0 \, u^2 \end{equation*}$

The first term here is the rest energy, which is constant for a body of fixed mass, while the second term is the Newtonian kinetic energy. Under these conditions, the relativistic expressions reduce to the classical expressions for energy and momentum, which explains why Newtonian mechanics provides a perfectly accurate description of motion at ordinary speeds.

As with any physical model, special relativity is valid within the domain defined by its assumptions. Within that domain, its predictions have been confirmed to extremely high precision in both laboratory experiments and operational engineering systems. In practice, special relativity serves as a kinematic framework that ensures the consistent conservation of energy, momentum, and time when interactions propagate at a finite, invariant speed. Newtonian mechanics emerges naturally as the low-speed, weak-field limit of general relativity.

What E = mc² really means

The equation $E = m\, c^2$ does not mean that mass is “turned into energy.” Rather, it states that mass itself carries energy because it carries inertia. The relation $E = m\,c^2$ is the rest-frame limit of the more general invariant energy–momentum relation, i.e.,

(135) $\begin{equation*} E^2 = (p\,c)^2 + (m_0 \, c^2)^2 \end{equation*}$

and the expression $E = m_0 \, c^2$ applies as a special case only when the momentum $p$ = 0. The appearance of the speed of light does not imply motion at speed $c$ ; instead, the factor $c^2$ serves as a universal conversion factor that places energy and inertia within a single, consistent bookkeeping framework. Newtonian mechanics deliberately neglects the rest energy $m_0 \, c^2$ because it is a constant term that does not affect forces, accelerations, or trajectories, just as an arbitrary reference level of potential energy is ignored in classical dynamics. Relativistic effects become apparent only when energy changes also change inertia, as occurs in systems involving high-speed radiation, relativistic particles, or precision timekeeping, where finite signal propagation prevents classical mechanics from remaining self-consistent with the conservation laws.

General Relativity

General relativity provides a geometric description of gravitation that remains consistent with finite signal propagation and non-inertial reference frames. In this theory, gravity is not treated as an instantaneous force acting at a distance. Instead, gravitational effects arise from the influence of mass and energy on the geometry of so-called spacetime.^[7] Objects, including light, follow trajectories determined by this geometry. One of the earliest experimental confirmations occurred during the 1919 solar eclipse, when starlight, a form of electromagnetic radiation, was observed to be deflected as it passed near the Sun.

From an engineering perspective, the need to consider general relativity arises primarily from problems that require precise timekeeping and reference-frame consistency, rather than from modifications to the equations of motion. For most aerospace applications, gravitational fields are weak, and velocities are small compared with the speed of light, so Newton’s law of gravitation provides an excellent approximation for forces and trajectories. Accordingly, relativistic corrections become relevant in systems in which minor timing errors can accumulate into measurable effects.

A prominent and practical application of relativity arises in satellite navigation systems. Clocks carried aboard navigation satellites do not remain synchronized with clocks on the Earth’s surface, and these differences must be corrected explicitly to preserve positional accuracy. The required corrections arise from two distinct effects that follow from consistent coordinate transformations between reference frames, rather than from any change in the physical operation of the clocks themselves.

The first contribution arises from gravitation and is obtained from the weak-field approximation for a spherically symmetric gravitating body. In this approximation, the rate at which time elapses for a clock located at radius $r$ relative to a clock far from the gravitating body is given by

(136) $\begin{equation*} \frac{\Delta t_r}{\Delta t_\infty} \approx \sqrt{1 - \frac{2 \, G \, M_E}{r \, c^2}} \end{equation*}$

where $G$ is the universal gravitational constant and $M_E$ is the mass of the Earth. This expression is not a fundamental field equation but an approximate result valid for weak gravitational fields and slowly moving clocks. Because the gravitational potential decreases with altitude, clocks located at higher orbital radii accumulate time at a slightly faster rate than clocks on the Earth’s surface. This effect is commonly referred to as gravitational time dilation, but it is more precisely interpreted as a difference in how elapsed time is assigned by coordinate systems associated with different gravitational potentials.

The second contribution arises from special relativity and is kinematic in origin. For a clock moving at speed $v$ relative to an inertial reference frame, the relationship between the elapsed time measured by the moving clock and the corresponding time interval measured in the inertial frame is

(137) $\begin{equation*} \Delta t' = \gamma \, \Delta t \qquad \text{with} \qquad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} \end{equation*}$

This effect follows directly from the Lorentz transformation of the time coordinate and is independent of gravitation. It reflects how time intervals transform between moving reference frames, not any physical slowing of the clock mechanism itself.

These relativistic effects are very real, and in satellite navigation systems such as the Global Positioning System (GPS), both gravitational and velocity-based corrections must be applied simultaneously. If these corrections are not included, satellite clocks drift relative to ground-based clocks, leading to rapidly accumulating position errors. This example illustrates why relativity enters aerospace engineering primarily through navigation, timing, and sensing systems, rather than through the routine equations governing vehicle forces and trajectories. The same considerations apply to deep-space probes, where relativistic corrections affect navigation and tracking via signal timing and reference-frame consistency rather than by modifying the vehicle’s equations of motion.

Check Your Understanding #9 – Relativistic effects with GPS

A navigation satellite has an orbital speed of 3.9 km/s. A ground clock is compared with an identical clock onboard the satellite. You are told that, for a GPS-class orbit, the gravitational (general relativity) contribution makes the satellite clock run faster by about 45 microseconds per day relative to the ground clock.

(a) Without calculation, state the sign of the velocity-based (special relativity) contribution to the satellite clock rate relative to the ground clock.
(b) Estimate the velocity-based time effect in microseconds per day using the approximation $\gamma - 1 \approx v^2/(2 \,c^2)$ for $v \ll c$ .
(c) Determine whether the satellite clock runs faster or slower overall and estimate the net time drift per day.
(d) Convert the net time drift into an approximate one-way range error, using 1 microsecond corresponds to about 300 m.

Show solution/hide solution.

(a) A moving clock runs slower than an identical clock at rest in the chosen inertial frame. Therefore, the velocity-based effect causes the satellite clock to run slower than the ground clock.

(b) Use the small-speed approximation

$\gamma - 1 \approx \frac{v^2}{2 \, c^2}$

Convert 3.9 km/s to SI units, giving $v = 3.9 \times 10^{3}$ . With $c = 3.0 \times 10^{8}$ , i.e.,

$\gamma - 1 \approx \frac{(3.9 \times 10^{3})^2}{2(3.0 \times 10^{8})^2} \approx 8.5 \times 10^{-11}$

So the fractional clock-rate decrease is about $8.5 \times 10^{-11}$ . Over one day (86,400 s), the time lost is approximately

$\Delta t \approx (8.5 \times 10^{-11})(86{,}400) \approx 7 \times 10^{-6}~\text{s}$

which is 7 microseconds per day.

(c) Combine the two effects. Gravitation makes the satellite clock run faster by about 45 microseconds per day, whereas motion makes it run slower by about 7 microseconds per day. Therefore, the net drift is about 38 microseconds per day, with the satellite clock running faster overall.

(d) Using 1 microsecond corresponds to about 300 m of one-way range, then

$\Delta R \approx 38 \times 300 \approx 11{,}400~\text{m}$

which is about 11 km per day.

The velocity-based effect makes the satellite clock run slower. For GPS-class orbits, the GPS clock drifts by about 7 microseconds per day; however, the gravitational effect is larger, so the satellite clock runs faster overall by about 38 microseconds per day, corresponding to a range error on the order of 10 km per day if uncorrected. Clearly significant!

Summary & Closure

The ability of engineers to apply the fundamental laws of physics is inseparable from their ability to solve real engineering problems. The principles reviewed in this chapter, conservation of mass, linear momentum, angular momentum, and energy, together with Newton’s laws of motion, gravitation, and electromagnetism, constitute the physical framework from which much of classical engineering analysis is constructed. These laws convert measured physical quantities into governing equations that describe, constrain, and ultimately predict the behavior of physical systems.

Every analysis encountered later in this eBook, whether in fluid mechanics, thermodynamics, propulsion, structures, or flight mechanics, is built upon these same foundations. They reappear in increasingly refined mathematical forms as the subject matter becomes more specialized and in ever more complex physical settings. Therefore, a rigorous conceptual and mathematical command of these physical laws is not merely helpful but essential for all advanced study and professional practice in engineering.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

Explain the physical meaning of a conservation law. Why are conservation principles so powerful in engineering analysis?
What is the fundamental difference between Newton’s first and second laws of motion?
Explain in your own words (don’t use the internet) the distinction between work and power.
Why is angular momentum conserved for motion under a central force?
What physical evidence demonstrates that electricity and magnetism are unified as a single interaction?
Explain in your own words (don’t use the internet) the distinction between mass and weight.

Other Useful Online Resources

There is much more to explore concerning the physical laws introduced in this chapter:

The Feynman Lectures on Physics, Volume I, provide a masterful conceptual treatment of classical mechanics and electromagnetism: https://www.feynmanlectures.caltech.edu
This video gives an intuitive overview of the major conservation laws in physics.
A concise review of Newton’s laws and their applications is provided in this Khan Academy unit.
This video series provides an accessible introduction to electricity and magnetism.

“The laws of nature are revealed through physics.” ↵
In the language of the seventeenth century, what is now called physics was known as natural philosophy. ↵
“Physics without mathematics is blind; mathematics without physics is empty.” Both are required for meaningful engineering analysis. ↵
Einstein first derived mass-energy equivalence in his 1905 paper analyzing the emission of electromagnetic radiation from a body at rest; see Einstein, A., ``Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?'' (``Does the Inertia of a Body Depend Upon Its Energy Content?''), Annalen der Physik, Vol.~18, pp.~639--641, 1905. ↵
Hendrik Lorentz developed it in the 1890s while trying to preserve Maxwell’s equations under coordinate transformations. ↵
The relativistic Doppler effect is the change in the observed frequency (and corresponding wavelength) of electromagnetic radiation because of relative motion between inertial observers, analogous to the classical Doppler effect with sound as proposed by Christian Doppler in 1842. ↵
Spacetime is used to provide a unified geometric framework for tracking position and time in systems where signals and interactions propagate at finite speed. ↵

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Introduction to Aerospace Flight Vehicles Copyright © 2022–2026 by J. Gordon Leishman is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, except where otherwise noted.

Introduction

Physical Laws in Engineering

Conservation Principles

Newton’s Absolutes

Newton’s Laws

Force, Mass, & Acceleration

Reference Frames & Coordinate Transformations

Translating Reference Frames

Rotating Reference Frames

The Free-Body Diagram

Work, Energy, & Conservation of Energy

Linear Momentum & Impulse

Impulse-Momentum Theorem

Conservation of Linear Momentum

Systems of Particles

Angular Momentum & Rotational Motion

Gravitation & Central-Force Motion

Newton’s Law of Universal Gravitation

Gravitational Field & Weight

Gravitational Potential Energy

Central-Force Motion

Equation of Motion in a Gravitational Field

Inverse-Square Law & Orbital Paths

Electricity & Magnetism

Electric Charge & Coulomb’s Law

Electric & Magnetic Fields

Electric Potential, Energy, & Current

Electromagnetic Induction

Maxwell’s Equations & Electromagnetic Fields

Energy in Electromagnetic Fields

Optics

Geometrical Optics & Ray Propagation

Radiative Transfer & Atmospheric Optics

Relativity Physics

Special Relativity

General Relativity

Summary & Closure

License

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