# Fundamentals of Physics: Mechanics, Relativity, and Thermodynamics

R. SHANKAR
Pages: 464
https://www.jstor.org/stable/j.ctt5vm4b6

1. Front Matter
(pp. i-vi)
(pp. vii-xii)
3. Preface
(pp. xiii-xiv)
4. CHAPTER 1 The Structure of Mechanics
(pp. 1-14)

This book is based on the first half of a year-long course that introduces you to all the major ideas in physics, starting from Galileo and Newton, right up to the big revolutions of the twentieth century: relativity and quantum mechanics. The target audience for this course and book is really very broad. In fact, I have always been surprised by the breadth of interests of my students. I don’t know what you are going to do later in life, so I have picked the topics that all of us in physics find fascinating. Some may not be useful, but...

5. CHAPTER 2 Motion in Higher Dimensions
(pp. 15-35)

In the last chapter we took the simplest case, of a point particle moving along thex-axis with a constant accelerationa. What is the fate of this particle? The answer is that at any timet, the location of the particle is given by

$x(t) = {x_0} + {v_0}t + \frac{1}{2}a{t^2}$, (2.1)

where${x_0}$and${v_0}$are its initial position and velocity. If you took the derivative of this, you would get

$v(t) = {v_0} + at$. (2.2)

You can easily check, by taking one more derivative, that this particle does indeed have a constant accelerationa. This equation, which gives the velocity of the object at timet,...

6. CHAPTER 3 Newton’s Laws I
(pp. 36-50)

This is a big day in your life: you are going to learn Newton’s laws, in terms of which you can understand and explain a very large number of phenomena. It’s really amazing that so much information can be condensed into three laws.

Your reaction may be that you have already seen Newton’s laws, that you have applied them in school. I realized fairly late in life that they are more subtle than I first imagined. It’s one thing to plug in all the numbers and say, “I know Newton’s laws and I know how they work.” But as you...

7. CHAPTER 4 Newton’s Laws II
(pp. 51-69)

The goal of physics is to be able to predict something about the future, given something about the present. I’m going to provide a very simple example that illustrates how Newton’s laws are to be used for this purpose. This treatment will be brief since I will return to this problem in greater detail later.

Figure 4.1 shows a frictionless table, on which is a massmattached to a spring of force constantk. The other end of the spring is attached to an immobile wall. The dotted outline of the mass shows it when it is displaced by...

8. CHAPTER 5 Law of Conservation of Energy
(pp. 70-81)

The law of conservation of energy is a robust and powerful one. When the laws of quantum mechanics were discovered in the subatomic world, many cherished notions were abandoned. You must have heard the ugly rumors: particles do not have a definite position and definite velocity at a given time. They don’t move along continuous trajectories. You might think the particle must have had an interpolating trajectory connecting two sightings, but it does not, and to assume it does causes conflict with experiment. While many of the ideas of Newtonian mechanics were abandoned, the notion of a conserved energy survived...

9. CHAPTER 6 Conservation of Energy in d = 2
(pp. 82-100)

We begin with some mathematical preparation for what I’m going to do next. Let’s take some function$f(x)$shown in Figure 6.1. I start at some pointxwith a value$f(x)$. When I go to a nearby point,$x + \Delta x$, the function changes by$\Delta f = f(x + \Delta x) - f(x)$. All these tiny quantities are exaggerated in the figure so you can see them. We are going to need approximations to the change in the function as$\Delta x \to 0$. A common one is to pretend the function is linear with the local value of the slope$f'(x) = \frac{{df}}{{dx}}$, as depicted by the dotted line. The change in...

10. CHAPTER 7 The Kepler Problem
(pp. 101-117)

Next we discuss one of the most famous problems involving a conservative force: celestial motion under the influence of Newtonian gravity. We’re going to make a big leap beyond inclined planes, pulleys, and whatnot; we are going to understand how planets move around the sun. That’s a mega problem, right? The littlem’s you put in the equation are not the masses of a pulley or a block, but the mass of Jupiter or the sun. You’re doing something of cosmological proportions. And you don’t need to know too much more to do that. You’re almost there.

The situation was...

11. CHAPTER 8 Multi-particle Dynamics
(pp. 118-142)

Next we begin our study of the dynamics of more than one body. You might think we already did this when we studied the solar system, consisting of the sun and all the planets. But we considered only one planet, and the sun just stood there as a source of the gravitational force. That was essentially a one-body problem.

As usual, let me start with the simplest possible case of two bodies moving in one dimension. They have coordinates${x_1}$and${x_2}$and masses${m_1}$and${m_2}$as shown in Figure 8.1. The first body obeys

${m_1}\frac{{{d^2}{x_1}}}{{d{t^2}}} = {F_1}$, (8.1)

Which I...

12. CHAPTER 9 Rotational Dynamics I
(pp. 143-158)

In this chapter we graduate to objects like potatoes that are not point-like. For such extended objects, it is not simple to say where “it” is. We can pick a point on the object, like the CM, and locate it, but still we do not have the whole story. We need to say which way the potato is pointing, an issue we did not have with point particles. We could go all the way and consider a body like a snake, which is not only extended but also capable of changing its shape. That is too hard, so I will...

13. CHAPTER 10 Rotational Dynamics II
(pp. 159-174)

Let us recall what we have learned about rigid bodies that are confined to lie and rotate in a plane, such as a rod or a sheet of some metal cut out into some arbitrary shape. The body has a massM. It can translate and rotate, but for now we nail a point on it to the plane and let it rotate about that axis, with plans to bring in translations later on. A single angle$\theta$, measured in radians, suffices to tell us what it is doing, because all it can do is rotate about the fixed point....

14. CHAPTER 11 Rotational Dynamics III
(pp. 175-193)

I’m going to consider cases where there is no external torque. If there’s no torque, we know the angular velocity is constant. But I’m going to take a case where this constant value of angular velocity is itself zero. There is no motion, there is no torque. So you might say, “What’s there to study?” Well, sometimes it’s of great interest to us to know that the object has no angular velocity, for example, if the object is a ladder we have climbed. The ladder better not have any angular acceleration either. What does it take to keep the ladder...

15. CHAPTER 12 Special Relativity I: The Lorentz Transformation
(pp. 194-208)

Although the general public associates the theory of relativity with Einstein’s monumental work of 1905, it is actually a lot older, going back to Galileo and Newton. According to the relativity principle, two observers in uniform relative motion will deduce the same laws of physics. That view of relativity has remained unchanged even after Einstein. However, in the Galilean version, the laws considered were those of mechanics, which was pretty much everything in those days. In the nineteenth century, it began to look as if the laws of electromagnetism and light did not respect the relativity principle. Einstein then rescued...

16. CHAPTER 13 Special Relativity II: Some Consequences
(pp. 209-226)

Let us begin with the Lorentz transformation, which relates the coordinates of an event in two different frames of reference, with the primed one moving at a velocityurelative to the unprimed one:

$x' = \frac{{x - ut}}{{\sqrt {1 - {u^2}/{c^2}} }}$(13.1)

$t' = \frac{{x - \frac{u}{{{c^2}}}x}}{{\sqrt {1 - {u^2}/{c^2}} }}$. (13.2)

The Lorentz transformation is the cornerstone of relativity; all the funny stuff you hear about—the shrinking rods, the twin paradox—everything comes from these simple equations, derived without even calculus. If you consider the stresses and strains on a loaded steel beam, the mathematics involved is a whole lot more difficult. That is the remarkable thing about relativity. In extracting the...

17. CHAPTER 14 Special Relativity III: Past, Present, and Future
(pp. 227-240)

In this chapter we continue to explore the consequences of the Lorentz transformation.

Let’s take the equation for the time difference between two events numbered 1 and 2:

$\Delta t' = \frac{{\Delta t - \frac{{u\Delta x}}{{{c^2}}}}}{{\sqrt {1 - \frac{{{v^2}}}{{{c^2}}}} }}$. (14.1)

First, something happens; then, something else happens, and$\Delta t = {t_2} - {t_1}$is a separation in time between them. Let’s say$\Delta t > 0$so that the second event occurred after the first event, according to me, using unprimed coordinates. How about according to you? Well,$\Delta t'$doesn’t have to have the same sign as$\Delta t$because you subtract from it this number,$\frac{{u\Delta x}}{{{c^2}}}$, which can be arbitrarily large and positive. Therefore, you can find...

18. CHAPTER 15 Four-momentum
(pp. 241-254)

In Newtonian mechanics particles have coordinates, let us say$(x,y)$, which could vary with time. From these we form a two-dimensional vector$r = ix + iy$. In a rotated frame the components$(x',y')$are given by

$x' = x\cos \theta + y\sin \theta$(15.1)

$y' = - x\sin \theta + y\cos \theta$. (15.2)

An entity V is defined as a vector (in two dimensions) if it has two components$({V_x},{V_y})$that, under rotation of the axes, go into${{V'}_x}$and${{V'}_y}$, related to$({V_x},{V_y})$exactly as in Eqn.15.1 and 15.2.

Now, I say to you, “Okay, that’s one vector, the position vector r. Can you point to another vector?” You might suggest the velocity$V = \frac{{dr}}{{dt}}$as...

19. CHAPTER 16 Mathematical Methods
(pp. 255-274)

I am going to introduce you to some mathematical tricks. As you’ve probably noticed by now, a lot of physics has to do with mathematics, and if you’re not good in math, you’re not going to be good in physics.

The first important trick is called the Taylor series. The philosophy of the Taylor series is the following: There is some functionf(x) depicted in Figure 16.1. But I’m going to imagine that you can only zero in on a tiny region nearx=0. And the question is, how will you write an approximation for this whole function, valid away...

20. CHAPTER 17 Simple Harmonic Motion
(pp. 275-302)

We’re now going to study what are called small oscillations, or simple harmonic motion. Take any mechanical system that is in a state of equilibrium. Equilibrium means the forces on the body add up to zero. It has no desire to move. If you give it a little kick, a push away from the equilibrium point, what will happen? There are two main possibilities. Imagine a marble on top of a hill. That is inunstable equilibriumbecause if you give the marble a nudge, it will roll downhill and never return to you. The other possibility involvesstable equilibrium:...

21. CHAPTER 18 Waves I
(pp. 303-315)

We are moving to another topic: waves. Everyone has a good intuitive feeling for waves. Suppose you drop some object in a lake and you see ripples traveling outward from the center. If you keep your eye level with the water you will find these ups and downs going outward. This is why one says a wave is some displacement of a medium. That’s not a perfect definition because electromagnetic waves travel in a vacuum. For this course you should imagine waves as what happens when you excite a medium. Once you have the example of water waves, you can...

22. CHAPTER 19 Waves II
(pp. 316-334)

It is obvious that a vibrating string has more energy than a string that is not vibrating. I want to calculate the energy in a string vibrating with displacement

$\psi (x,t) = A\cos (kx - \omega t)$. (19.1)

Now, if it’s an infinitely long string, the energy in it is infinite, so you define the energy per unit length. Take a portion of the string, associated with a segment of widthdx, and ask, “ How much energy does it have?”

The energy has a kinetic part and a potential part. The kinetic part is simple:

$dk = \frac{1}{2}\mu dx\left[ {\frac{{\partial \psi }}{{\partial t}}} \right]$. (19.2)

This is just the mass of the segment times...

23. CHAPTER 20 Fluids
(pp. 335-351)

This is a relatively simple topic. If you took any kind of high school physics, you would have seen fluids. Whenever I say fluid, you are free to imagine water or oil.

Let us begin with a basic property of the fluid, the density, denoted by$\rho$. The density of water is${\rho _w} = 1,000{\rm{ }}kg/{m^3}$. The more subtle concept is the one ofpressure. If you dive down to the bottom of a swimming pool, the pressure goes up. What is the formal definition of pressure? That’s what I want to explain.

If we pick a point in the fluid and say...

24. CHAPTER 21 Heat
(pp. 352-374)

This chapter—devoted to the study of heat, temperature, and heat transfer—sets the stage for our study of thermodynamics.

You already have an intuitive notion of temperature. Let us begin here with what may be new: the notion ofthermal equilibrium. Systems are said to be in thermal equilibrium when theirmacroscopicproperties, properties discernible by macroscopic probes like the naked eye or a thermometer, have stopped changing.

Take a cup of hot black coffee and take another cup of cold pink soda, and keep them both thermally isolated from each other and the outside world. No matter how...

25. CHAPTER 22 Thermodynamics I
(pp. 375-393)

In the last chapter we took the notion of temperature, for which we have an intuitive feeling, and turned it into something more quantitative, so you can not only say this is hotter than that, you can say by how much, by how many degrees. We agreed to use the absolute Kelvin scale for temperature and to use the product ofPVof a gas thermometer as a measure of temperatureT. The Kelvin scale has its origin (T= 0) at —273.16°C, which is wherePVvanished for any dilute gas. In other words, it appears that pressure times...

26. CHAPTER 23 Thermodynamics II
(pp. 394-410)

Let us begin with the first law

$dU = \Delta Q - \Delta W = \Delta Q - PdV$, (23.1)

which says that$dU$, the change in the energy of the gas, equals the heat input$\Delta Q$minus the work donebythe system,$\Delta W = PdV$. Why do we refer to some infinitesimals with a$\Delta$and some with ad?

This has to do with whether these refer to simply small quantities or to small quantities that correspond to a change in astate variable, which is some function that depends on the state of the gas, specified byPandV. Consider the internal energyU. It is a state...

27. CHAPTER 24 Entropy and Irreversibility
(pp. 411-442)

As promised at the end of the last chapter, we will now go from Carnot’s practical considerations on the efficiency of heat engines to the notion of entropy.

In computing the efficiency of the Carnot engine, we found that the heat absorbed from the reservoir at${T_1}$and the heat rejected into the reservoir at${T_2}$are in the ratio (consult Figure 23.4)

$\frac{{{Q_2}}}{{{Q_1}}} = \frac{{{T_2}}}{{{T_1}}}$, (24.1)

which we can rewrite as

$\frac{{{Q_1}}}{{{T_1}}} - \frac{{{Q_2}}}{{{T_2}}} = 0$. (24.2)

Note that in this convention,${Q_1}$was the heatabsorbedfrom the hot reservoir duringAB, while${Q_2}$was the heatrejected in CDinto the cold reservoir....

28. Index
(pp. 443-446)