WJEC Physics for A2: Study and Rev Guide

Vertical oscillations of an object hanging from spring The equation a = − w 2 x in which w 2 = k m also applies to the vertical oscillations of m in Fig. 3.2.3. This is far easier to set up than the arrangement in Fig. 3.2.1, but note that x , the displacement from equilibrium, is not now the spring extension. Why not? 3.2.2 Variation of displacement with time for a body in shm As we’ll con rm later, a body will have acceleration given by a = − w 2 x if (and only if) its displacement varies with time according to x = A cos ( w t + e ) ( w , A and e are constants.) This motion is closely related to that of a point, P , moving with angular velocity w round and round a circle of radius A . In fact, x is the horizontal component of P ’s displacement from the circle centre (Fig. 3.2.4), and w t and e are the angles shown. A x ε ω t P at t = t P at t = 0 A x 0 0 −A T time 2 T t Fig. 3.2.4 x = A cos ( w t + e ) as circular motion component A is the amplitude of the shm: the maximum displacement. It is a constant determined by how the body is set in motion, for example by how far we displace it from its equilibrium position, before letting go. The periodic time , T , the time for one cycle of oscillation, is the time for one revolution of point P – see Fig. 3.2.4. So, from Section 3.1.2, w = 2 p T that is T = 2 p w . Equivalently, as frequency , f = 1 T we have f = w 2 p that is w = 2 p f . ■ For a mass–spring system, since w = k m , we have T = 2 p m k m x EQ’M fixed support m DISPLACED Fig. 3.2.3 Vertical oscillations The amplitude of an oscillation is the maximum value of the displacement. The periodic time (or period), T , is the time for one cycle. The frequency , f , is the number of cycles per unit time. UNIT: s −1 = hertz (Hz) . The phase , ( w t + e ), is the angle that tells us the point reached in the oscillation cycle. Key Terms 14 A Level Physics: Study and Revision Guide