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Newtonian Mechanics in one dimension As explained in the last section, we consider a body which can move freely along a horizontal track fixed to the surface of the earth. The track is rigid and straight, and has tick marks one meter apart. −4 −3 −2 −1 0 1 2 3 4 We also have a uniformly-running clock, with which we can measure time in seconds. Using the clock and the tick marks on the track, we can determine the velocity of the body. For present purposes, all we really need is to be able to say whether the velocity is constant or not. The principle of inertia Suppose we start the body with some initial velocity. Then eventually it slows down and stops due to friction. However, we can repeat the experiment using ever more elaborate methods of reducing friction. We find that it becomes harder and harder to detect the decrease in velocity. By suitably refining the apparatus, it is possible to come arbitrarily close to the ideal situation in which the velocity of the body remains constant. According to Newton, this is no accident, but rather a fundamental law of nature. This law is often called Newton's first law, but a more descriptive name is The principle of inertia: • (tentative version only!) In the ideal case in which a body is not acted upon by any external influence, its velocity will remain constant. Inertia is the name given to this tendency of a body to remain in a state of uniform motion unless acted on by an external influence. Two remarks are needed, in order to make the above statement of the principle of inertia precise. First, the body must have constant mass. The meaning of "constant mass" should be intuitively clear, although we have not yet actually defined mass. Second, it is important to specify what we are measuring the velocity with respect to. The principle of inertia is not valid, for example, if we measure the velocity of our freelymoving body with respect to a track which is accelerating - speeding up or slowing down - with respect to the surface of the earth. To see this, suppose the body and track are initially at rest with respect to the surface of the earth. Then the track begins to speed up to the right. As seen from the earth, the body remains at rest - moves with a constant velocity of zero: (This is what happens when a waiter quickly pulls a tablecloth out from under some dishes.) From the point of view of the track, however, the body begins to speed up to the left: Its velocity with respect to the track is therefore not constant. Similarly, if the clock does not run uniformly, then the velocity of the body with respect to the fixed track will not appear to be constant, even if no external influence acts on the body. The track, together with the clock, is our frame of reference. If the track is not accelerating and the clock runs uniformly, then the principle of inertia is valid and the frame is an inertial frame. If not, then the frame is noninertial. To be accurate, the last part of the above statement of the principle of inertia should read ...its velocity with respect to an inertial frame will remain constant. A final comment is in order. In our example of a track on the earth, we have ignored two effects: the rotation of the earth, and tidal forces. These cause the surface of the earth to be a noninertial frame. We will deal with these phenomena in later chapters. For present purposes, it is sufficient to know that their effects can be made negligibly small in the experiment described. Newton's second law Now let's see what the effect of an "external influence" is. We will join the body to a spring whose other end is attached to the track, and then stretch the spring. If we hold the body at rest and then release it, it accelerates to the right - its velocity increases. We say that a force has been applied to the body. Force: • Anything which causes a body having constant mass to accelerate relative to an inertial frame is called a force. Recall that we have units for time and distance seconds and meters, respectively. Notice that the above definition does not tell us what units to use for force, or how to measure its value in those units. This definition is qualitative, rather than quantitative. To make it quantitative, we need to give a precise definition of mass. Suppose we now join a second, identical body to the first one. We stretch the spring by the same amount as before, and observe the motion of the combined bodies. We look for a relation between the motion of the original body and that of the combined bodies. Of all the properties of the motion, one in particular is related very simply. We find that the acceleration is exactly half as large as before. (We are talking here about the initial acceleration, when the springs are still both stretched by the same amounts.) Notice that it didn't have to be this way. The acceleration could have been some other fraction of its original value. Or the property which behaves simply could have been the rate of change of the acceleration. There are many such possibilities. But the motion happens to be governed by a law of nature which says that the acceleration is half as large for the combined bodies, for a given force. Now, the size of the initial force applied to the combined bodies is the same as that applied to the single body. The force is something external to the body, its value being completely determined by the internal properties of the spring and by how far the spring was stretched. And the spring was stretched by the same initial amount in both cases. What has changed is the amount of matter contained within the body - this has doubled. The word for the "amount of matter", used in this sense, is mass. The mass of the new body is twice that of the old single body. If we made another new body out of three of the original ones, the mass would be three times as large. The measured initial acceleration would be one-third as large as the original. Thus, the acceleration produced by a given force is inversely proportional to the mass. What is the unit of mass? All we can do is pick some reference object and say that it has one unit of mass. Then the masses of all other bodies are fixed in terms of this unit. We simply compare their accelerations under the influence of the same force. By convention, we choose the mass unit to be the kilogram. Note that this procedure is no different from the choice of second and meter as the units of time and distance. The choice is completely arbitrary. (In this course, we will not be concerned with the actual standards which define the second, meter and kilogram.) Newton's second law: Now let's consider what happens when we double the force, but leave the mass the same. We do this by using two identical springs instead of one, and stretching them by the same amount as before. • In an inertial frame, We compare the motion with the original, and find that the initial acceleration is doubled. Again, this did not have to be so, but nature says it is. For three springs, the acceleration is tripled, and so on. The acceleration is directly proportional to the force, when the mass is constant. If the acceleration is inversely proportional to the mass when the force is constant, and directly proportional to the force when the mass is constant, then the mathematical relation between them is acceleration = (a number) H force ÷ mass. We fix the number by choosing the size of the unit of force. We say that a force of one unit acting on a mass of one unit produces an acceleration of one unit. Then the number is 1. It is conventional to multiply both sides of the above relation by the mass, and express the relation symbolically in the form of F = ma for a body having constant mass. The unit of force is symbolized by the letter N, and is called the Newton: 1 N = 1 kg m s–2 (kg stands for kilogram, m for meter and s for second. The unit of acceleration is m s–2 - read "meters per second per second".) As we have defined it, mass is a measure of a body's resistance to being accelerated. The larger the mass of a body, the less it will be accelerated by a given force. It has "more inertia", so to speak. For this reason, it is appropriate to call mass defined in this way inertial mass. When we later study gravity, we will see that there is another kind of mass, the gravitational mass, defined completely differently. The distinction between the two definitions of mass is tremendously important, as is the experimental fact that they are one and the same! Re-stated using the concept of force, the principle of inertia reads The principle of inertia: that a force acts on the body. But this force is fictitious - it is entirely an effect of the acceleration of the reference frame. • In the ideal case in which a body having constant mass is not acted upon by any external force, its velocity with respect to an inertial frame will remain constant. Despite the fact that such forces are fictitious, they are extremely important and we will have occasion to discuss them in detail later on. It is interesting to look at the relation between Newton's second law and the principle of inertia. The principle of inertia deals with the case in which the force on the body is zero, while Newton's second law tells what happens when the force is arbitrary. So the principle of inertia is just a particular case of Newton's second law the case where the force is zero. We should check that this is reflected in the mathematics. Beginning with F=ma, we set F to zero. The mass is not zero, so the acceleration a must be zero. But zero acceleration is the same as constant velocity. This shows that Newton's second law reduces to the principle of inertia when the external force is zero. We note that Newton's second law is not valid in a noninertial frame. We return to our earlier example in which the track accelerates but the body remains at rest with respect to the earth. From the point of view of the track, the body accelerates. A person attached to the track might then apply Newton's second law and conclude Several forces More than one distinct force can act on a body. For example, a body could have several springs attached to it, and there could be friction present, and so on. In this case, it is found experimentally that the body’s acceleration is the same as that which would result from the application of a single force which is the sum of the individual forces. In symbols, ma = F net = 3 F i , i where the index i labels the individual forces. The sum of all the individual forces is sometimes called the “net” force. In this course, we will often write “F=ma” with the understanding that F means the net force if more than one force is acting. Newton's second law in action In many cases, the nature of the net force acting on a body is known. It might depend on time, position, velocity, or some combination of these, but its dependence is known from experiment. In such cases, Newton's law becomes an equation of motion which we can solve. How is this equation obtained? We begin with Newton’s law, F=ma, where the (net) force is a function of x, v, and t: F(x,v,t) . Using the definition of the velocity and acceleration as the first and second derivatives of x(t), respectively, we get the equation of motion äå d 2 x(t) dx(t) ëìì å m = F x ( t ) , ,t ì . å dt í dt2 ã This is a differential equation for the function x(t). That is, it is an equation whose solution is a whole function of time, not just an algebraic number. The solution allows us to predict the position of the body at any time, as long as we know its initial position and velocity. This predictive quality is the main power of Newton’s law. Don’t worry if this looks a bit abstract right now. In the next few sections, we will see how it works in several concrete cases. Here is a directory of the cases we will consider: • constant force, with and without damping • force proportional to distance (the harmonic oscillator), with and without damping • (advanced topic) vertical motion near earth, taking into account variation of gravity with height.