If we take the lowest point of the pendulum (y 0) as the reference of U, then, making the same substitutions as above.
Harmonic Oscillator Animation Movie Clip ShowsThis movie clip shows a torsional wave first a travelling wave, then a standing wave.The track is perforated with small holes, through which flows air from the inside, where the pressure is above atmospheric. Consequently, the only non-negigible force in the horizontal direction is that exerted by the two springs. Because there are no vertical displacements, we discuss here only the horizontal displacement. The point at which it stops is, of course, its maximum displacement to the left. The spring force now acts to the left, so it decelerates until it stops at its maximum rightwards displacement. In this case, k k 1 k 2, where k 1 and k 2 are the constants of the two springs. However, we do a quantitative analysis on the multimedia chapter Oscillations and also solve this problem as an example on Differential Equations. This is explained in detail in the Kinematics of Simple Harmonic Motion in Physclips. The sine function goes through one complete cycle when its argument increases by 2, so we require that ((tT) ) (t ) 2, so T 2, so. For a linear system, the frequency is independent of amplitude (see below, however, a for nonlinear system ). The first uses one air track glider and the second uses two similar gliders, so the mass is doubled. The period is increased by about 40, i.e. Here the period is shorter and therefore the frequency higher that in all the previous examples. This is because of the air track: here, no non-negligible nonconservative forces act, so mechanical energy is conserved. The horizontal displacement of m from the position of equilibrium is x, and the string makes an angle with the vertical, as shown in the sketch. For the moment, we consider only the case in which is small, so x. Further, cos, which may be expanded as cos 1 2., is approximately 1. So the magnitude T of the tension in the string is approximately mg. Substituting this in the preceding equation and rearranging gives. Writing T as the period (not to be confused with the magnitude of the tension T ), we write 1T f 2 (12)(gL).
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