Figure 5.2: Illustrating that the force Fg of a jet airplane on hot gases is equal in magnitude and opposite in direction to the force Faof the gases on the airplane.
forces between two bodies which are touching each other. However, the third law is only approximate for forces, like gravitational and electrical forces, which act between two separated bodies. If the bod-ies are moving rapidly with respect to each other, an improved de-scription is given by Einstein’s theory of relativity, which we discuss in Chapter 10.
5.4 Frames of reference
As far as we know, Newtons’s first law (the law of inertia) is ex-act. However, we know that Newton’s second and third laws are only approximate, although they hold very well for slowly-moving macroscopic bodies. By slow, we mean slow compared to the speed
of light, which is very fast indeed. An airplane moves fast compared to a person walking, but the airplane moves very slowly compared to the speed of light. A jet plane in normal flight moves at about 900 kilometers per hour, while light moves at about 300,000 kilometers per second (186,000 miles per second). The second and third laws are so good for slowly-moving bodies that in many cases we do not have instruments sensitve enough to measure any deviations from them.
We digress here to introduce a notation for very large and very small numbers, as we encounter them often in this book. The speed of light (300,000 km/s) can also be written as 3 × 105 km/s. The notation 105 means 1 followed by 5 zeros, or 100,000. Similarly, 108 means 1 followed by 8 zeros or 100,000,000 (one hundred million).
We use negative powers of 10 for very small numbers. For example, the number 10−4 means 1 divided by 10,000, or 0.0001,
In discussing Newton’s laws, we have to point out a certain lim-itation in their validity that has nothing to do with speed. Let us concentrate our attention on the law of inertia, which is exact inde-pendent of speed. Nevertheless, there is an important limitation to the law of inertia in that it is observed to hold only when measure-ments are made from a so-called inertial frame of reference.
What is a frame of reference, and what is an inertial frame? Before defining these notions, we illustrate them with an example. Go out-side on a clear night and look up at the stars. Even though they may be moving rapidly, they seem motionless because they are very far away. That is why they are often called the “fixed” stars. However, suppose you rotate your body while looking at the stars. Then, if you
5.4. FRAMES OF REFERENCE 65 consider yourself at rest, the stars make circles overhead–they are no longer motionless from your point of view. If I remain at rest and watch both you and the stars, then, from my point of view the stars are fixed (not moving) and you are rotating. Who is to say whose point of view is “better,” yours or mine?
A “point of view,” used in this sense, is called a “frame of ref-erence.” Sometimes, we shorten this simply to “frame.” According to modern ideas, both your frame of reference and mine are equally good; this is a principle of general relativity, proposed by Einstein and discussed in Chapter 10.
It turns out that Newton’s first law (the law of inertia) holds for a person making a measurement if that person is in the frame of refer-ence in which the stars are fixed (motionless). For that reason, such a frame of reference is called an “inertial frame.” Furthermore, it turns out that any frame of reference which is moving at constant velocity with respect to the fixed stars is also an inertial frame. However, a frame of reference that is accelerating with respect to the frame of the stars is not an inertial frame. If you turn in a circle, you are changing your direction, which means you are accelerating, and so you are not in an inertial frame.
Even to a person who is stationary on earth, the stars do not ap-pear to be absolutely at rest. This is because the earth rotates on its axis once a day, so the stars make apparent circles in the sky once a day. Because the motion is not in a straight line, it is accelerated mo-tion. The apparent rotation of the stars shows that the frame of the earth is not strictly an inertial frame. This implies that an observer who is stationary on earth is not strictly in an inertial frame.
How-ever, because the earth is large and rotates on its axis only once a day, the earth is very nearly in an inertial frame. In fact, for many mea-surements, we can neglect the fact that the earth is rotating and use Newton’s laws of motion without modification.
There are some measurements, however, that clearly show that the earth’s frame is rotating. For example, consider a pendulum sus-pended from a high ceiling by a wire. If you give the pendulum a push, it will swing back and forth in a straight line. Make a chalk line on the floor parallel to the direction the pendulum is swinging.
If you leave the room for a few hours and then return, you will find that the pendulum no longer swings parallel to the chalk line, but makes an angle with respect to it. The force that causes the direction of the swing to change is a “fictitious” or “inertial” force. From the point of view of an observer in an inertial frame (not rotating with the earth), the pendulum does not change its direction of swing at all, but the earth rotates underneath it.
Fortunately, physicists are able to take account of the earth’s ro-tation and make more precise calculations than would be possible if the earth’s frame is taken as an inertial frame. Nevertheless, unless we state otherwise, we shall regard the earth’s frame of reference as an inertial frame. In doing so, we are making an approximation, and our results will have an error. Physicists frequently make approxi-mations to simplify calculations. A physicist usually has a good idea of the size and importance of the errors he makes as a result of his approximations. If the errors are unimportant, the approximations are useful.
Physicists occasionally make approximations even if they know
5.4. FRAMES OF REFERENCE 67 the resulting errors will be big, or if they don’t know how important the errors will be. Physicists may make such approximations either because they will be satisfied with a rough estimate of the value of a quantity or because they cannot solve the problem exactly. The philosophy in the latter case is that a poor answer is better than no answer at all. (This is not always true.)
There are also errors in measurements, so even when a physicist measures a quantity, he obtains only an approximate answer. When we say a calculation (prediction) from a theory and the result of a measurement ”agree,” we mean only within the errors of the mea-surement and the calculation.
The “rest frame” of an object is the frame in which it is at rest.
An observer makes observations and measurements from his own rest frame, no matter how he may be moving with respect to an-other frame, such as the frame in which the earth is stationary or the frame in which the stars are fixed. The rest frame of an object may or may not be an inertial frame, depending on how the object is moving when viewed from an inertial frame.
It has been observed that the frame in which the stars are fixed is an inertial frame and that all frames of reference which move with constant velocity with respect to the frame of the fixed stars are also inertial frames. Because of this, we cannot use the law of inertia to determine whether we are at absolutely at rest or whether we are moving at constant velocity. From the point of the law of inertia, absolute motion at constant velocity is not observable, and we can measure only relative motion. However, there is a way to measure the local rest frame of the universe. We cannot measure a global rest
frame of the universe because the universe is expanding, as we shall discuss in Chapter 19.
If you ride in a car at constant velocity, you do not feel any forces on you. Therefore, if you close your eyes, you cannot tell how fast you are moving or even whether you are moving at all. (Actually, you can tell that you are moving because no car can travel with strictly constant velocity. Small forces cause the car to vibrate when it is moving, and vibration is accelerated motion.) If you open your eyes and look out the window, you see the trees at the side of the road rushing backwards. Because you know that the trees are sta-tionary on the earth, you know you are moving with respect to the earth, or, alternatively, that the earth is moving with respect to you.
We next consider accelerating frames of reference. By an accel-erating frame, we mean a frame which is acclaccel-erating with respect to the fixed stars. A rotating frame is a special case of an accelerating frame.
It is easy to see that the law of inertia does not hold in accelerating frames. Slide a smooth stone over a smooth floor, so that friction can be neglected (to a good approximation). In this approximation, there are no forces acting on the stone and so it slides with constant ve-locity, in accordance with the law of inertia. However, while you are watching the stone, accelerate your own body. If you are accelerat-ing with respect to the stone, the stone is accelerataccelerat-ing with respect to you in the opposite direction. Therefore, When measured from your frame, the stone accelerates even though no forces are acting on it, in violation of the law of inertia. Thus, the law of inertia is violated when measured from an accelerating (noninertial) frame.
5.4. FRAMES OF REFERENCE 69 It is apparant from this example that not only does Newton’s first law fail to hold in an accelerating frame, but also his second law.
After all, the first law is a special case of the second, so that if the first law fails, so does the second. The third law, however, holds to a good approximation in accelerating frames, especially for contact forces.
Newton’s first two laws can be modified to hold in an accelerating frame only by introducing so-called “fictitious” forces to account for the acceleration of objects measured with respect to the accelerating frame.
As an example of a “fictitious” force arising in an accelerating frame, consider what happens when you turn to the left in a car while traveling at high speed. (This is an acceleration by virtue of a change in direction.) The answer is that you will feel yourself pushed to the right, and if you don’t wear your seat belt, you may actually slide along the seat to the right. Although the force that pushes on you to the right seems real enough to you, it is regarded as a “fictitious”
force from the point of view of an observer in an inertial frame. Ac-cording to the observer in the inertial frame, the car turns to the left because of a force on it in the left direction arising from the friction of the road on the wheels. According to the same observer, you are also pushed to the left by the seat belt which is attached to the car.
The observer cannot identify any force on you to the right. If you say you feel a force to the right, he will say it is a fictitious force be-cause you are not in an inertial frame. It is your inertia that resists the force pushing you to the left and feels like a force pushing you to the right. Therefore, sometimes the “fictitious” forces are called
“inertial” forces. Neither word for the extra force felt by someone in
a non-inertial frame is a good one. The term “fictitious” is not good for a force that feels as real as any other force. The term “inertial” is not good because the force is absent when measured from an iner-tial frame of reference. But “fictitious” and “ineriner-tial” are the words commonly used, and so we use them in this book.
The conclusion of all this is that observers making measurements from accelerating frames need to include forces which seem real enough from their own point of view but are absent from the point of view of an observer making measurements from an inertial frame.
Chapter 6
Newton’s Theory of Gravity
If I have seen farther than others, it is because I have stood on the shoulders of giants.
—Isaac Newton
6.1 An apple and the moon
Newton showed that the motion of an apple falling to earth and the motion of the moon around the earth arise from the same force of gravity. In order to understand how the motion of an apple in a straight line and the motion of the moon in an approximate circle (actually an ellipse) can arise from the same force law, we first exam-ine motion on earth in some detail.
Gravity is the force that makes objects fall to earth. But these ob-jects do not always fall in a straight line. It is true that an object dropped from rest, like the apple, will fall straight down. However, if you throw a ball horizontally, its subsequent motion in the air will be a combination of the horizontal motion you originally gave it and
71
vertical motion due to gravity. The actual path is curved, and is ap-proximately given by the curve called a parabola. in Figure 6.1 we illustrate the path of a ball thrown initially with horizontal velocity.
Figure 6.1: Parabolic path of a ball thrown with initial horizontal velocity. Pictures of the ball are given at equal time intervals.
The horizontal component of the ball’s velocity will decrease slightly because of air resistance, while the vertical component of the veloc-ity will increase because of the downward force of gravveloc-ity. The net result is that the ball starts out horizontally, but falls at a steeper and steeper angle from the horizontal until it hits the ground.
Imagine throwing the ball horizontally in the absence of air re-sistance. The horizontal component of the velocity will remain con-stant while the vertical component increases because of gravity until the ball hits the ground. Then imagine throwing the ball at a greater initial speed. It will of course travel farther before it hits the ground.
One can conceive of throwing the ball at such a high speed that it never hits the ground (neglecting irregularities in the earth such as mountains). The ball curves toward the earth, but the earth is also
6.1. AN APPLE AND THE MOON 73 curved so that a sufficiently fast-moving ball can remain a constant distance above the surface of the earth even as it continues to fall.
This is approximately the situation with the moon, except that it or-bits the earth in an ellipse rather than in a circle. The moon keeps falling toward the earth, but the horizontal component of its velocity is sufficiently large that it it remains in orbit above the surface of the earth.
Newton guessed that the force of gravity between two pointlike objects is proportional to the product of their masses and inversely proportional to the square of the distance between them. It is an attractive force. The constant of proportionality is called the “gravi-tational constant” and is denoted by the symbol G. In symbols, the magnitude of the force is given by F = Gm1m2/r2,where m1and m2
are the masses of the two objects and r is the distance between them.
Newton was not the only one to guess the inverse square law, but he showed that the law was universal, acting on the moon and the apple with the same coupling constant G. In order to show this, he had to assume that the earth attracts the apple as if all the earth’s mass were concentrated at its center. Later he proved this assump-tion by taking the earth to be a sphere and summing up the gravita-tional contributions from every bit of matter in the sphere.
Newton did not know the mass of the earth or moon, and he did not know the value of the gravitational constant G. However, he assumed the same value of G and the same velue of the mass of the earth in both cases, so that by taking the ratio of the forces (dividing one by the other) the quantities G and the mass of the earth cancel out and so do not need to be known.
Newton also did not need to know the mass either of the moon or of the apple because of his second law of motion, which says that the acceleration of a body is inversely proportional to its mass. Because the acceleration is obtained by dividing the force by the mass and because the gravitational force is proportional to the mass, the body’s mass cancels out in the expression for the acceleration.
The net result of all of this is that the ratio of the acceleration of the moon to the acceleration of the apple equals the inverse ratio of the squares of their distances to the center of the earth. In order to calculate the ratio, Newton had to know the radius of the earth and the distance to the moon, numbers that he was well aware of. He then compared his result with measurements of the accelerations of the moon and the apple, and found that they agreed with his calcu-lation.
The fact that acceleration of a body due to gravity is independent of its mass is why Galileo found that a large stone and a small stone fall with the same acceleration.