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Newtonian Mechanics We are now going to proceed with Newtonian mechanics in three dimensions. The treatment will parallel that given in the earlier section on Newtonian mechanics in one dimension, and will be more brief. An inertial frame typically consists of a uniformly-running clock and a set of identical measuring rods with uniform tick marks, mounted at right angles to one another. The rods are not accelerating in any way (in particular, not rotating). In more than one dimension, position, velocity, and acceleration are vectors: xP, vP, and a P . If the position is a function of time, then the velocity and acceleration are vP = z 2 dxP dvP d xP and Pa = = . dt dt dt2 The first step is to state the principle of inertia and use it to define what we mean by an inertial frame. The principle of inertia • If a body having constant mass is not acted upon by any external influence, its velocity will remain constant. Inertial frames • An inertial frame is one in which the principle of inertia holds. ä y x We will choose to measure time in seconds and distance in meters. Now that we have defined inertial frames, and know how to tell when no external influence is acting, we can proceed to cases in which an external influence does act. of the same external influence. By convention, we choose the mass unit to be the kilogram. Mass The next step is to define (inertial) mass. We do this by first figuring out a way to apply the same external influence to different bodies in an inertial frame. For example, we stretch a spring by a fixed amount, and attach composite bodies consisting of varying numbers of identical bodies to the spring. fixed In this case, we know that the external influence doesn’t depend on the nature of the body on which it acts. It is determined solely by the distance the spring is stretched. We find that the acceleration of a composite body is inversely proportional to the number of its constituents - that is, to the m a s s of the composite body. We 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 Newton's second law We then figure out how to vary the external influence, but leave the mass the same. For example, we do this by using several identical springs instead of one, and stretching them by the same amount as before. fixed We find that the acceleration is proportional to the number of springs - that is, to the force exerted by the bunch of springs. 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 takes the form of Newton's second law: Newton's second law in action • In an inertial frame, P = m aP F for a body having constant mass. Notice that this is a vector equation. Both the force and the acceleration have a direction associated with them, and Newton’s second law says that their directions are the same. In cases in which more than one distinct force acts on a single body, 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, P = 3F P , ma P=F net i i where the index i labels the individual forces. The sum of all the individual forces is sometimes called the “net” force. In many cases, the nature of the 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 a set of three equations of motion which we can solve. There is one equation for each dimension. Example: motion under a constant force As our first example, we consider the motion of a P - one body under the action of a constant force F which does not vary over time, is the same no matter where the body is, and does not depend on the body's velocity. We begin with Newton’s law. Using the definition of the acceleration as the first derivative of the velocity, we find the following three equations of motion: P = m dvP(t) . F dt Rearranging and integrating once, we find vP(t) = P F t + a constant vector, m where the constant vector does not depend on time. The constant must be fixed by the value of the velocity at some particular time. For simplicity, let's suppose that this time is t=0, and that the velocity at that time is vP0 . Substituting t=0 into both sides, we find that the value of the constant vector is vP0 . The velocity is therefore given by P dxP(t) F = t + vP0 . dt m We then integrate again, and find xP(t) = P 2 1 F t + Pv0 t + another constant vector. 2 m The new constant vector is fixed by the value of the position at another particular time, which we may again take to be t =0. If the position then is xP0 , then the value of the new constant is xP0 . The final answer for the position as a function of time is thus given by xP(t) = P 2 F t + Pv0 t + xP0 2m . This formula allows us to find the position at any time, as long as we know the values of the initial position and velocity vectors. The reason we have to know two vectors is because Newton's law gives rise to a second-order vector differential equation. That is, the highest derivative which appears is the second derivative. You might find it useful to compare this analysis with our previous treatment of motion under a constant force in one dimension. You will find that they are remarkably similar.