So at last we have solved the equation that we really wanted to solve. The equation d=x/dr=-wdx is the same as Eq(21.2)if wo=k/ The next thing we must investigate is the physical significance of wo. We know that the cosine function repeats itself when the angle it refers to is 2. So x= cos wot will repeat its motion, it will go through a complete cycle, when the angle"changes by 2. The quantity wof is often called the phase of the motion In order to change wot by 2, the time must change by an amount tb, called the period of one complete oscillation; of course to must be such that wofo=2T That is, wofo must account for one cycle of the angle, and then everything will repeat itself-if we increase t by fo, we add 2T to the phase.Thus to=2丌/ao=2m√m/k Thus if we had a heavier mass, it would take longer to oscillate back and forth on a spring. That is because it has more inertia, and so, while the forces are the same it takes longer to get the mass moving. Or, if the spring is stronger, it will move more quickly, and that is right: the period is less if the spring is stronger Note that the period of oscillation of a mass on a spring does not depend in any way on how it has been started, how far down we pull it. The period is deter mined, but the amplitude of the oscillation is not determined by the equation of motion(21. 2). The amplitude is determined, in fact, by how we let go of it, by what we call the initial conditions or starting conditions Actually, we have not quite found the most general possible solution of (21.2). There are other solutions. It should be clear why: because all of the cases covered by x a cos wot start with an initial displacement and no initial velocity. ut it is possible, for instance, for the mass to start at x=0, and we may then give it an impulsive kick, so that it has peed at (= 0. Such a motion is not represented by a cosine--it is represented by a sine. To put it another way, if x= cos wof is a solution, then is it not obvious that if we were to happen to walk nto the room at some time(which we would call"t=0")and saw the mass as it was passing x =0, it would keep on going just the same? Therefore, x =cos wot cannot be the most general solution; it must be possible to shift the beginning of time, so to speak. As an example, we could write the solution this way: x a cos wo(t -f1), where t1 is some constant. This also corresponds to shifting the origin of time to some new instant. Furthermore, we may expand cos(cot+△)= coS wot cos△- sin wot sin△ and write x= A coS wot Bsin wot cos 4 and B=-asin 4. Any one of these forms is a possible way to write the complete, general solution of (21. 2): that is, every solution of the differential equation dx/dr= -wax that exists in the world can be written as (a) x= acos wo(t-41) (b)x=acos(aot+△), = A cos wot+Bsinωot Some of the quantities in(21. 6)have names: wo is called the angular frequency is the number of radians by which the phase changes in a second. That is deter- mined by the differential equation. The other constants are not determined by the equation, but by how the motion is started. Of these constants, a measures the maximum displacement attained by the mass, and is called the amplitude of oscilla tion. The constant A is sometimes called the phase of the oscillation, but that is a confusion, because other people call wot A the phase, and say the phase changes with time, We might say that A is a phase shift from some defined zero. Let us put it differently. Different 4's correspond to motions in different phases. That is true, but whether we want to call A the phase, or not, is another question 21-3