# Volume A: Kinetics, Statics, and Thermodynamics

- Page ID
- 6218

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 1A: Mathematical Prelude
- One of your goals in taking a physics course is to become more proficient at solving physics problems, both conceptual problems involving little to no math, and problems involving some mathematics. In a typical physics problem you are given a description about something that is taking place in the universe and you are supposed to figure out and write something very specific about what happens as a result of what is taking place.

- 2A: Conservation of Mechanical Energy I: Kinetic Energy & Gravitational Potential Energy
- Physics professors often assign conservation of energy problems that, in terms of mathematical complexity, are very easy, to make sure that students can demonstrate that they know what is going on and can reason through the problem in a correct manner, without having to spend much time on the mathematics. A good before-and-after-picture correctly depicting the configuration and state of motion at each of two well-chosen instants in time is crucial in showing the appropriate understanding. A pres

- 3A: Conservation of Mechanical Energy II: Springs, Rotational Kinetic Energy
- A common mistake involving springs is using the length of a stretched spring when the amount of stretch is called for. Given the length of a stretched spring, you have to subtract off the length of that same spring when it is neither stretched nor compressed to get the amount of stretch.

- 4A: Conservation of Momentum
- A common mistake involving conservation of momentum crops up in the case of totally inelastic collisions of two objects, the kind of collision in which the two colliding objects stick together and move off as one. The mistake is to use conservation of mechanical energy rather than conservation of momentum. One way to recognize that some mechanical energy is converted to other forms is to imagine a spring to be in between the two colliding objects such that the objects compress the spring.

- 5A: Conservation of Angular Momentum
- The angular momentum of an object is a measure of how difficult it is to stop that object from spinning. For an object rotating about a fixed axis, the angular momentum depends on how fast the object is spinning, and on the object's rotational inertia (also known as moment of inertia) with respect to that axis.

- 6A: One-Dimensional Motion (Motion Along a Line): Definitions and Mathematics
- A mistake that is often made in linear motion problems involving acceleration, is using the velocity at the end of a time interval as if it was valid for the entire time interval. The mistake crops up in constant acceleration problems when folks try to use the definition of average velocity in the solution. Unless you are asked specifically about average velocity, you will never need to use this equation to solve a physics problem. Avoid using this equation—it will only get you

- 7A: One-Dimensional Motion: The Constant Acceleration Equations
- The constant acceleration equations presented in this chapter are only applicable to situations in which the acceleration is constant. The most common mistake involving the constant acceleration equations is using them when the acceleration is changing.

- 8A: One-Dimensional Motion: Collision Type II
- A common mistake one often sees in incorrect solutions to collision type two problems is using a different coordinate system for each of the two objects. It is tempting to use the position of object 11 at time 00 as the origin for the coordinate system for object 11 and the position of object 22 at time 00 as the origin for the coordinate system for object 22 . This is a mistake. One should choose a single origin and use it for both particles. (One should also choose a single positive

- 9A: One-Dimensional Motion Graphs
- Consider an object undergoing motion along a straight-line path, where the motion is characterized by a few consecutive time intervals during each of which the acceleration is constant but typically at a different constant value than it is for the adjacent specified time intervals. The acceleration undergoes abrupt changes in value at the end of each specified time interval. The abrupt change leads to a jump discontinuity in the Acceleration vs. Time Graph and a discontinuity in the slope.

- 10A: Constant Acceleration Problems in Two Dimensions
- In solving problems involving constant acceleration in two dimensions, the most common mistake is probably mixing the x and y motion. One should do an analysis of the x motion and a separate analysis of the y motion. The only variable common to both the x and the y motion is the time. Note that if the initial velocity is in a direction that is along neither axis, one must first break up the initial velocity into its components.

- 11A: Relative Velocity
- Vectors add like vectors, not like numbers. Except in that very special case in which the vectors you are adding lie along one and the same line, you can’t just add the magnitudes of the vectors.

- 12A: Gravitational Force Near the Surface of the Earth, First Brush with Newton’s 2nd Law
- Some folks think that every object near the surface of the earth has an acceleration of 9.8 m/s downward relative to the surface of the earth. That just isn’t so. In fact, as I look around the room in which I write this sentence, all the objects I see have zero acceleration relative to the surface of the earth. Only when it is in freefall, that is, only when nothing is touching or pushing or pulling on the object except for the gravitational field of the earth, will an object experience

- 13A: Freefall, a.k.a. Projectile Motion
- The constant acceleration equations apply from the first instant in time after the projectile leaves the launcher to the last instant in time before the projectile hits something, such as the ground. Once the projectile makes contact with the ground, the ground exerts a huge force on the projectile causing a drastic change in the acceleration of the projectile over a very short period of time until, in the case of a projectile that doesn’t bounce, both the acceleration and the velocity become ze

- 14A: Newton’s Laws #1: Using Free Body Diagrams
- If you throw a rock upward in the presence of another person, and you ask that other person what keeps the rock going upward, after it leaves your hand but before it reaches its greatest height, that person may incorrectly tell you that the force of the person’s hand keeps it going. This illustrates the common misconception that force is something that is given to the rock by the hand and that the rock “has” while it is in the air. It is not. A force is all about something that is being done to

- 15A: Newton’s Laws #2: Kinds of Forces, Creating Free Body Diagrams
- There is no “force of motion” acting on an object. Once you have the force or forces exerted on the object by everything (including any force-per-mass field at the location of the object) that is touching the object, you have all the forces. Do not add a bogus “force of motion” to your free body diagram. It is especially tempting to add a bogus force when there are no actual forces in the direction in which an object is going. Keep in mind, however, that an object does not need a force on it to

- 16A: Newton’s Laws #3: Components, Friction, Ramps, Pulleys, and Strings
- When, in the case of a tilted coordinate system, you break up the gravitational force vector into its component vectors, make sure the gravitational force vector itself forms the hypotenuse of the right triangle in your vector component diagram. All too often, folks draw one of the components of the gravitational force vector in such a manner that it is bigger than the gravitational force vector it is supposed to be a component of.

- 17A: The Universal Law of Gravitation
- Consider an object released from rest an entire moon’s diameter above the surface of the moon. Suppose you are asked to calculate the speed with which the object hits the moon. This problem typifies the kind of problem in which students use the universal law of gravitation to get the force exerted on the object by the gravitational field of the moon, and then mistakenly use one or more of the constant acceleration equations to get the final velocity.

- 18A: Circular Motion - Centripetal Acceleration
- There is a tendency to believe that if an object is moving at constant speed then it has no acceleration. This is indeed true in the case of an object moving along a straight line path. On the other hand, a particle moving on a curved path is accelerating whether the speed is changing or not. Velocity has both magnitude and direction. In the case of a particle moving on a curved path, the direction of the velocity is continually changing, and thus the particle has acceleration.

- 19A: Rotational Motion Variables, Tangential Acceleration, Constant Angular Acceleration
- One of the most common mistakes we humans tend to make is simply not to recognize that when someone asks us; starting from time zero, how many revolutions, or equivalently how many turns or rotations an object makes; that someone is asking for the value of the angular displacement Δθ . To be sure, we typically calculate ΔθΔθ in radians, so we have to convert the result to revolutions before reporting the final answer, but the number of revolutions is simply the value of Δθ.

- 20A: Torque & Circular Motion
- The mistake that crops up in the application of Newton’s 2nd Law for Rotational Motion involves the replacement of the sum of the torques about some particular axis, ∑τ , with a sum of terms that are not all torques. Oftentimes, the errant sum will include forces with no moment arms (a force times a moment arm is a torque, but a force by itself is not a torque) and in other cases the errant sum will include a term consisting of a torque times a moment arm.

- 21A: Vectors - The Cross Product & Torque
- Do not use your left hand when applying either the right-hand rule for the cross product of two vectors (discussed in this chapter) or the right-hand rule for “something curly something straight” discussed in the preceding chapter.

- 22A: Center of Mass, Moment of Inertia
- A mistake that crops up in the calculation of moments of inertia, involves the Parallel Axis Theorem. The mistake is to interchange the moment of inertia of the axis through the center of mass, with the one parallel to that, when applying the Parallel Axis Theorem. Recognizing that the subscript “CM” in the parallel axis theorem stands for “center of mass” will help one avoid this mistake. Also, a check on the answer, to make sure that the value of the moment of inertia with respect to the axis

- 23A: Statics
- It bears repeating: Make sure that any force that enters the torque equilibrium equation is multiplied by a moment arm, and that any pure torque that enters the torque equilibrium equation is NOT multiplied by a moment arm.

- 24A: Work and Energy
- You have done quite a bit of problem solving using energy concepts. Back in chapter 2 we defined energy as a transferable physical quantity that an object can be said to have and we said that if one transfers energy to a material particle that is initially at rest, that particle acquires a speed which is an indicator of how much energy was transferred. We said that an object can have energy because it is moving (kinetic energy), or due to its position relative to some other object (potential ene

- 25A: Potential Energy, Conservation of Energy, Power
- The work done on a particle by a force acting on it as that particle moves from point A to point B under the influence of that force, for some forces, does not depend on the path followed by the particle. For such a force there is an easy way to calculate the work done on the particle as it moves from point A to point B.

- 26A: Impulse and Momentum
- Imagine a gigantic air hockey table with a whole bunch of pucks of various masses, none of which experiences any friction with the horizontal surface of the table. Assume air resistance to be negligible. Now suppose that you come up and give each puck a shove, where the kind of shove that you give the first one is special in that the whole time you are pushing on that puck, the force has one and the same value; and the shove that you give each of the other pucks is similar in the following respe

- 27A: Oscillations: Introduction, Mass on a Spring
- When something goes back and forth we say it vibrates or oscillates. In many cases oscillations involve an object whose position as a function of time is well characterized by the sine or cosine function of the product of a constant and elapsed time. Such motion is referred to as sinusoidal oscillation. It is also referred to as simple harmonic motion.

- 28A: Oscillations: The Simple Pendulum, Energy in Simple Harmonic Motion
- Starting with the pendulum bob at its highest position on one side, the period of oscillations is the time it takes for the bob to swing all the way to its highest position on the other side and back again. Don’t forget that part about “and back again.”

- 29A: Waves: Characteristics, Types, Energy
- Consider a long taut horizontal string of great length. Suppose one end is in the hand of a person and the other is fixed to an immobile object. Now suppose that the person moves her hand up and down. The person causes her hand, and her end of the string, to oscillate up and down.

- 30A: Wave Function, Interference, Standing Waves
- In that two of our five senses (sight and sound) depend on our ability to sense and interpret waves, and in that waves are ubiquitous, waves are of immense importance to human beings. Waves in physical media conform to a wave equation that can be derived from Newton’s Second Law of motion. The wave equation reads:

- 31A: Strings, Air Columns
- Be careful not to jump to any conclusions about the wavelength of a standing wave. Folks will do a nice job drawing a graph of Displacement vs. Position Along the Medium and then interpret it incorrectly. For instance, look at the diagram on this page.

- 32A: Beats and the Doppler Effect
- If a single frequency sound source is coming at you at constant speed, the pitch (frequency) you hear is higher than the frequency of the source. How much higher depends on how fast the source is coming at you. Folks make the mistake of thinking that the pitch gets higher as the source approaches the receiver. That would be the case if the frequency depended on how close the source was to the receiver. It doesn’t. The frequency stays the same. The Doppler Effect is about velocity, not position.

- 33A: Fluids: Pressure, Density, Archimedes' Principle
- One mistake you see in solutions to submerged-object static fluid problems, is the inclusion, in the free body diagram for the problem, in addition to the buoyant force

- 34A: Pascal’s Principle, the Continuity Equation, and Bernoulli’s Principle
- Experimentally, we find that if you increase the pressure by some given amount at one location in a fluid, the pressure increases by that same amount everywhere in the fluid. This experimental result is known as Pascal’s Principle.

- 35A: Temperature, Internal Energy, Heat and Specific Heat Capacity
- As you know, temperature is a measure of how hot something is. Rub two sticks together and you will notice that the temperature of each increases. You did work on the sticks and their temperature increased.

- 36A: Heat: Phase Changes
- As mentioned in the preceding chapter, there are times when you bring a hot object into contact with a cooler sample, that heat flows from the hot object to the cooler sample, but the temperature of the cooler sample does not increase, even though no heat flows out of the cooler sample (e.g. into an even colder object).

- 37A: The First Law of Thermodynamics
- We end this physics textbook as we began the physics part of it (Chapter 1 was a mathematics review), with a discussion of conservation of energy. Back in Chapter 2, the focus was on the conservation of mechanical energy; here we focus our attention on thermal energy.

Thumbnail: Roller coaster "Blue Fire" at Europa Park. (CC SA 3.0; Coaster J).