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Useful Formulae

34 SIMPLEHARMONICMOTION vibrating and oscillating phenomena Galileo discovered that the periodTof a pendulum, the time it takes to swing from one side to the other and back again, is independent of its amplitude,its maximum displacement from the centre. Thus a pendulum of length l completes f swings every second,regardless of whether its swings are wide or narrow. Here f is the frequencyof the pendulum. If l is given in metres,the periodTis given in seconds by T 2 where g is the gravitational constant see page 28. For small swings,less than 5,pendula approximate simple harmonic motion. Bobbing bottles and sproinging springs all perform s.h.m., as do a vast array of vibrating and oscillating phenomena. Potential and kinetic energies are continually exchanged. The potential energy and acceleration of the vibrating object reach maxima when its kinetic energy and velocity are minimal and vice versa. A simple way to generate s.h.m.is to project a uniform circular motion onto an axis as shown opposite,giving dt a sint where ais the amplitude and the angular velocity. The period is then seconds per cycle and the frequency is the reciprocal of the period, cycles per second. f 1 gl 2 2 35
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