Perhaps the most appealing feature of classical mechanics is its logical economy. Everything is derived from Newton’s three laws of motion. (Well, almost everything. One must also know something about the forces which are acting.) It is necessary of course to understand quite clearly what the laws assert and to acquire some experience in applying the laws to specific situations.
We are concerned here with the first and third laws. The second law will only be discussed in passing here. Time spent in thinking about the meaning of these laws is (to put it mildly) not wasted.
The first law, in Newton’s own words, is as follows: “Every body perseveres in its state of resting, or uniformly moving in a right line, unless it is compelled to change that state by forces impressed upon it”1. In modern language, the first law states that the velocity of a body is constant if and only if there are no forces acting on the body or if the (vector) sum of the forces acting on the body is zero. Note that when we say the velocity is constant, we mean that both the magnitude and direction of the velocity vector are constant. In this statement we assume that all parts of the body have the same velocity. Otherwise, at this early point in the discussion we do not know what we mean by “the velocity of a body.”
Two questions immediately arise:
The answers to (a) and (b) are related. In fact, if we are willing to introduce a sufficiently complicated notion of force, the first law will be true with respect to every set of axes and says nothing. The “sufficiently complicated notion of force” would involve postulating that whenever we see the velocity of a particle changing a force is acting on the particle even if we cannot see the source of the force.
We shall insist on giving the word “force” a very restricted meaning which corresponds closely to the way the word is used in everyday language. We define a force as the push or pull exerted by one piece of matter on another piece of matter. This definition is not quantitative (a quantitative measure of force will be introduced shortly) but emphasizes the fact that we are entitled to speak of a “force” only when we can identify the piece of matter which is exerting the force and the piece of matter on which the force is being exerted.
Some simple examples will illustrate what we mean and do not mean when we use the word “force.”
We shall see that the relatively simple notion of “force” which we have defined is quite sufficient for our purposes. Many forces are encountered in everyday life, but if we look closely enough they can all be explained in terms of the gravitational attraction exerted by one piece of matter (usually the earth) on another, and the electric and magnetic forces exerted by one charged body on another. We shall frequently refer to the “contact” forces exerted by one body on another when their surfaces are touching. These “contact” forces can, in general, have a component perpendicular to the surface and a component parallel to the surface; the two components are called, respectively, the normal force and the frictional force. If we examine the microscopic origin of these forces, we find that they are electric forces between the surface molecules (or atoms) of one body and the surface molecules of the other body. Even though the molecules have no net charge, each molecule contains both positive and negative charges; when two molecules are close enough, the forces among the various charges do not exactly cancel out and there is a net force. Fortunately, the application of Newton’s laws does not require a detailed microscopic understanding of such things as contact forces; nevertheless we refuse to include any force on our list unless we are convinced that it can ultimately be explained in terms of gravitational, electric, and/or magnetic forces.3
Now that we know what we mean by a force, we can ask “With respect to which axes is it true that a particle subject to no forces moves with constant velocity?”; that is, with respect to which axes is Newton’s first law true? A set of axes is frequently called a “frame of reference,” and those axes with respect to which Newton’s first law is true are called inertial frames.
It is important to note that, as a consequence of Newton’s first law, there is more than one inertial frame. If a set of axes are an inertial frame, and if another set of axes are moving with constant velocity and not rotating with respect to , then are also an inertial frame. This follows from the fact that a particle which has constant velocity with respect to the axes will also have a constant velocity with respect to the axes.
We have already seen that axes attached to an accelerating railroad car are not an inertial frame, but axes attached to the earth are an inertial frame. Actually, this is not quite true. For most purposes, axes attached to the earth appear to be an inertial frame. However, due to the earth’s rotation, these axes are rotating relative to the background of the distant stars. If you give a hockey puck a large velocity directed due south on a perfectly smooth ice rink in Philadelphia, it will not travel in a perfectly straight line relative to axes scratched on the ice, but it will curve slightly to the west because of the earth’s rotation. This effect is important in naval gunnery and illustrates that axes attached to the earth are not a perfect inertial frame. A better, but less convenient, set of axes are axes which are nonrotating with respect to the distant stars and whose origin moves with the center of the earth.
Another phenomenon which demonstrates that axes attached to the earth’s surface are not a perfect inertial frame is the Foucault pendulum. A plumb bob attached by a string to the ceiling of a building at the North or South Pole will oscillate in a plane which is nonrotating with respect to the distant stars. The earth rotates relative to the plane of the pendulum.
Newton was principally interested in calculating the orbits of the planets. For this purpose he used axes whose origin is at rest with respect to the sun, and which are nonrotating with respect to the distant stars. These are the best inertial frame he could find and he seems to have regarded it as obvious that these axes are at rest in “absolute space” which “without relation to anything external, remains always similar and immovable.” Kepler observed that, relative to these axes, the planets move in elliptical orbits and that the periodic times of the planets (i.e., the time required for the planet to make one circuit around the sun) are proportional to the 3/2 power of the semimajor axis of the ellipse. Newton used his laws of mechanics, plus his law of universal gravitation (which gave a quantitative formula for the force exerted by the sun on the planets), to explain Kepler’s observations, assuming that the axes in question are an inertial frame or a close approximation thereto. Furthermore, he was able to calculate the orbit of Halley’s comet with great accuracy.
Most (perhaps all) physicists today would say that the notion of “absolute space” is elusive or meaningless, and that the inertial frames are physically defined by the influence of distant matter. Ordinarily, when we enumerate the forces acting on a body we include only the forces exerted on it by other bodies which are fairly close to it; for example, if the body is a planet we take account of the gravitational force which the sun exerts on the planet but we do not explicitly take account of the force which the distant stars exert on the planet (nor do we really know how to calculate that force). Nevertheless, the effect of the distant stars cannot be ignored since they determine which are the inertial frames; that is, they single out the preferred axes with respect to which Newton’s laws are true. The important characteristic of axes attached to the sun is not that they are at rest in absolute space, but that they are freely falling under the gravitational influence of all the matter outside the solar system.
In most of the homely examples which we shall discuss, axes attached to the surface of the earth can be treated as an inertial frame. In discussion of planetary motion we shall use axes attached to the sun which do not rotate relative to the distant stars.
Newton’s second law, states that the acceleration of a body is proportional to the total force acting on the body. Some authors use this fact as the basis for a quantitative definition of force. We propose to define “force” quantitatively before discussing the second law. Thus it will be clear that the second law is a statement about the world and not just a definition of the word “force.” We shall then gain familiarity with the analysis of forces by studying a number of examples of the static equilibrium of particles.
As the unit of force we choose some elementary, reproducible push or pull. This could, for example, be exerted by a standard spring stretched by a standard amount at a standard temperature. Two units of force would then be the force exerted by two such standard springs attached to the same object and pulling in the same direction (depending on the characteristics of the spring this might or might not be the same as the force exerted by a single spring stretched by twice the standard amount).
More whimsically, if we imagine that we have a supply of identical mice which always pull as hard as they can, then the “mouse” can be used as the unit of force. One mouse pulling in a given direction can be represented by an arrow of unit length pointing in that direction. Three mice pulling in the same direction can be represented by an arrow of length three pointing in that direction. The definition is easily extended to fractional numbers of mice; for example, if it is found that seven squirrels pulling in a given direction produce exactly the same effect as nineteen mice pulling in that direction, then we represent the force exerted by one squirrel by an arrow of length 19/7 in the appropriate direction. Thus any push or pull in a definite direction can be represented by an arrow in that direction, the length of the arrow being the number of mice required to perfectly mimic the given push or pull.
Since we have established a procedure for representing forces by arrows which have lengths and directions, it seems almost obvious that forces have all the properties of vectors. In particular, suppose two teams of mice are attached to the same point on the same body. Let one team consist of mice all pulling in the same direction (represented by a vector ), and let the other team consist of mice all pulling in another direction (represented by a vector ).
We should clearly understand that when we represent forces by vectors we are saying not only that a force has magnitude and direction, but that two forces (each represented by a vector) acting simultaneously at the same point are equivalent to a single force, represented by the vector sum of the two force vectors. It follows that when more than two forces act at a point, they are equivalent to a single force, represented by the vector sum of the vectors representing the individual forces.
Now we can discuss the equilibrium of point masses. A “point mass” is a body so small that we measure only its location and ignore the fact that different parts of the body may have different velocities. We shall see shortly that, as a consequence of Newton’s third law, Newton’s laws apply not only to point masses but to larger composite objects consisting of several or many point masses as well.
A body is said to be in equilibrium when it is at rest (not just for an instant, but permanently or at least for a finite period of time4) or moving with constant velocity. According to Newton’s first law, there is no force on a body in equilibrium.
The simplest example of equilibrium would be a particle in the outer reaches of the solar system, sufficiently far from the sun and the planets so as to be subject to negligible gravitational forces. This example is not very interesting since no forces act on the particle. The more interesting examples of equilibrium, as encountered in everyday life, are situations in which the net (or “resultant”) force on a body is zero, but several forces are acting on the body; thus the equilibrium of the body results from the fact that the vector sum of all the forces acting on the body is zero.
As a first example of equilibrium, we consider a block at rest on the floor. We assume here that the block can be treated as a “point mass” which obeys Newton’s first law. Even if the block is not very small we shall see shortly that Newton’s third law justifies treating the block as a “point mass.”
Two forces act on the block: the earth pulls downward on the block and the floor pushes upward on the block. Since the total force on the block must vanish, the two forces must be equal in magnitude and opposite in direction. However they are entirely different in their origins. The downward pull exerted by the earth (frequently called “gravity” or the “weight” in colloquial terms) is simply the vector sum of the gravitational forces exerted on the block by every molecule of the earth. The contribution to this sum from nearby molecules (within a few miles from the block) is negligible; the gravitational force on the block is caused mainly by its attraction to distant molecules because there are so many distant molecules. On the other hand, the upward force exerted by the floor is a very short-range electrical force (the “contact” force which we have already mentioned) exerted by molecules located in the surface of the floor on molecules located in the bottom of the block.
The magnitude of the gravitational force exerted by the earth on a body is called the weight of the body and is usually denoted by the letter . The numerical value of depends on what we choose as the unit of force. In the so-called English system of units the unit of force is not the mouse but the pound. The pound may be defined as the gravitational force exerted by the earth on a certain standard object placed at a certain location on the earth’s surface. For example, the object could be 454 cubic centimeters (27.70 cubic inches) of water at a temperature of 4 degrees centigrade and atmospheric pressure, located at 33rd and Walnut Streets in Philadelphia. In the metric system the unit of force is the newton, which is approximately 0.225 pounds. (Note: the Google web site will do many kinds of common conversions for you. There is no need to memorize specific conversion factors. Nevertheless, it is useful to know (at least approximately) common conversion factors, e.g., meters feet, centimeters inches, kilowatts horsepower, etc.) We often abbreviate the newton unit as N. It is often useful to draw a free-body diagram in which each of the forces acting on a body is represented by a vector. Usually all of the vectors are drawn originating from a common origin. If the body is in equilibrium, the vector sum of all the forces in the free-body diagram is zero. In the present example, the free-body diagram is very simple (Figure 2) and adds nothing to our verbal description. In the following more interesting examples, quantitative conclusions can be drawn from the free-body diagram.
Consider a block in equilibrium on a smooth inclined plane. The earth pulls downward on the block and the plane exerts a contact force on the block in a direction normal (perpendicular) to the plane. Evidently, a third force must be applied to the block if it is to be in equilibrium. Experience tells us that the direction of this third force is not uniquely determined. For example, the block can be kept in equilibrium by an appropriate force applied parallel to the plane in the uphill direction (Figure 3) or by an appropriate horizontal force (Figure 4). In fact, there is a continuum of possible directions for the third force. If the block is to be in equilibrium, the third force must have the appropriate magnitude. This depends on the direction in which it is applied.
Consider a block suspended by a pair of strings of equal length both of which are attached to the ceiling and make an angle with the horizontal (see Figure 5). The free-body diagram exhibits the forces acting on the block. is the gravitational force exerted by the earth and and are the forces exerted by the two strings. From the symmetry of the problem it is evident that and have the same magnitude, which we call . If we take the -axis horizontal and the -axis vertical, the components of the vector equation are The first of these equations tells us nothing but merely confirms our assumption that the tensions in both strings are equal. The second tells us that the required tension in the strings is .
Note that becomes very large when is very small (in fact as ). Thus we see that a rather modest sideways force applied to a taut wire can break the wire. However, if we apply a sideways force to the center of a taut nylon rope, the rope will stretch; since does not remain small, the tension in the rope will not become very large.
In the previous example the two strings might have been of different lengths, so that (Figure 6). Note that the two strings must be separately attached to the block. (If the block hangs from a ring which is free to slide along the string, the ring will slide until ) In the present case and the components of the force equation are Solving this pair of simultaneous linear equations, we find Note that when the result agrees with that of the previous example.
We have not yet considered the static equilibrium of systems which consist of several bodies, which may exert forces on each other. In order to discuss such systems we must introduce a very important property of forces which has not yet been mentioned and cannot be deduced from anything which has been said up to this point. Newton’s statement of this property is the following:
“To every action there is always opposed an equal reaction; or, the mutual actions of two bodies upon each other are always equal, and directed to contrary points.”
This is called Newton’s Third Law of Motion. Newton goes on to give some examples of the third law:
“If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may say so) will be equally drawn back towards the stone…”
In modern language, the third law can be stated as follows:
“For every force which exerts on , exerts an equal and oppositely directed force on ”
These forces are called an action-reaction pair.
The third law has very important consequences and it is essential to understand exactly what the third law asserts. Let us look again at Section 4.1 (the block on the floor). There are two forces acting on the block: (the gravitational force exerted by the earth) and (the contact force exerted by the floor). The reaction to is the gravitational force exerted on the earth by the block. This force is represented by the vector ; that is, the magnitude of the gravitational force exerted by the block on the earth is but the direction of this force is upward. The reaction to is the contact force exerted by the block on the floor. As we have previously noted, this force is a very short-range electrical force. This force has the same magnitude as but with opposite direction (downward) and is represented by the vector .
A common misconception is that and are an action-reaction pair. Note that the two forces in an action-reaction pair do not act on the same body (one is exerted by on while the other is exerted by on ). Since and act on the same body (the block), they cannot be an action-reaction pair. Furthermore, the two forces in an action-reaction pair are of the same physical origin; for example, both are gravitational forces or both are contact forces. But is a gravitational force and is a contact force, so again we see that they cannot be an action-reaction pair.
The third law is true even when the bodies under consideration are not in equilibrium. For example, a body falling toward the earth exerts a gravitational force on the earth which is equal and opposite to the gravitational force exerted by the earth on the body. When a baseball bat strikes a ball, the ball exerts a force on the bat which is equal in magnitude and opposite in direction to the force which the bat exerts on the ball. A familiar paradox is posed by the question “If the force exerted by the ball on the bat is equal and opposite to the force exerted by the bat on the ball, then why does the ball accelerate?” The answer is provided by Newton’s second law, which states that the acceleration of the ball is proportional to the net force acting on the ball. Thus, the force exerted by the ball on the bat is irrelevant to the question of whether the ball accelerates.
The temptation to (incorrectly) identify and as an action-reaction pair arises partially from the fact that in equilibrium . If we consider a nonequilibrium situation in which the block is on the floor of an elevator which is accelerating upward, then the net force acting on the block is not zero; that is, . In this case the block accelerates upward because the upward contact force exerted by the floor on the block is greater than the downward gravitational force exerted on the block by the earth. Nevertheless, Newton’s third law is valid: the gravitational force exerted on the earth by the block is equal and opposite to the gravitational force exerted on the block by the earth, and the contact force exerted on the floor by the block is equal in magnitude and opposite in direction to the contact force exerted on the block by the floor.
Strictly speaking, not all the forces in nature obey the third law exactly.5 However, if we stay away (as we will in this book) from situations in which a large amount of electromagnetic radiation is being produced, the third law is true. A commonly (and erroneously) cited example of a force which violates the third law is the magnetic force between two short segments of wire, each of which carries an electric current. In fact, the force between two current-carrying segments is not measurable, and the only physically significant force is the force between two closed loops of wire. This obeys the third law.
As a first illustration of the use of Newton’s third law we consider a slight generalization of the example we studied in Section 4.2. Suppose we have two blocks on a smooth inclined plane, the upper block (#2) being supported by a lower block (#1), which is supported by an external force applied parallel to the incline.
The free-body diagram for block #2 (see Figure 7) includes the force exerted by the earth, exerted by the plane, and the force which block #1 exerts on block #2. We assume that the blocks are rectangular with smooth faces so that is parallel to the incline. The free-body diagram for block #1 includes (in addition to and ) the force which block #2 exerts on block #1. By Newton’s third law, this force is . Furthermore the free-body diagram for block #1 must include the external force which is applied to block #1. Note clearly that is a force acting on block #1, not on block #2. The force exerted by #1 on #2 is .
The condition of equilibrium for block #2 is
and for block #1
The components of Equation (3a) and Equation (3b) along axes parallel and perpendicular to the incline are
These are four equations in the four unknowns , , , . Solving, we obtain
Note that without the third law we would have to introduce another unknown force (the force which block #2 exerts on block #1, which we might call ). We would then have four equations in five unknowns and the problem would not be mathematically determinate.
In the example above, the formulae for and can be obtained directly from Equation (2a) and Equation (2b) since the problem of the equilibrium of the upper block is identical with the problem we have already solved in Section 4.2. More important is the observation that the formula Equation (3d) for can be obtained directly from Equation (2a) if we think of the two blocks as one composite object of weight . Similarly, from Equation (2b), the total normal force on this composite object is .
Is it always permissible to say that the total force on an object in equilibrium is zero, even when the object is a composite system consisting of several parts? With the aid of the third law we can prove that the answer is “yes.” Were this not the case, Newton’s first law would be applicable only to certain “elementary” objects (presumably of microscopic dimensions) and could not be applied to objects like balls and blocks which really consist of many molecules.
In short, the third law permits us to sidestep the delicate question of what are the “particles” which obey Newton’s laws of motion. If sufficiently small objects obey Newton’s first law, then the third law implies that larger objects will also obey the first law.6
We define a system as any collection of particles (a particle is an object sufficiently small that it obeys Newton’s laws). The particles are enumerated by an index . We say that the system is in equilibrium when every particle of the system is in equilibrium (i.e., at rest or moving with constant velocity).
If is the total force acting on the th particle, then in equilibrium for every , and thus We can write as the sum of two terms where is the external force on the th particle (i.e., the force exerted on the th particle by particles which are not included in the system) and is the force exerted by the th particle on the th particle.7 Thus the equation becomes