WJEC Physics for A2: Study and Rev Guide

■ For a simple pendulum (small bob hanging by a light thread from a xed support), swinging no further than about 20° from the vertical, the bob’s acceleration is related to its displacement x along the arc by a = − w 2 x ■ in which w = g l , so T = 2 p l g . ( w t + e ) is an angle called the phase . It tells us the stage reached in the oscillation cycle at time t . e is an angle called the phase constant. We can set its value so that time zero is at any point we choose in the body’s cycle... If we want t = 0 to be when x = A, we choose e = 0 , giving x = A cos( w t ) . (Fig. 3.2.5 (a)) If we want t = 0 to be when x = 0 with x about to be positive we choose e = − p 2 . Because cos w t − p 2 = sin ( w t ), we have x = A sin( w t ) . (Fig. 3.2.5 (b)) Example 1 A metal sphere of mass 0.16kg hangs from a spring of spring constant 40Nm −1 . The sphere is released from rest with a displacement of +0.030m . Calculate the period of its oscillations and its displacement 0.55 s after release. Answer w = k m = 40Nm −1 0.16kg = 15.8 [rad] s −1 So T = 2 p w = 2 p 15.8 s −1 = 0.40 s . A = 0.030m , because the displacement will oscillate between ±0.30m . e = 0 if we call the release time (from maximum displacement) t = 0 . So x = A cos ( w t ) = 0.030m × cos(15.8 × 0.55 rad) = −0.022m . ■ To evaluate cos(15.8 × 0.55 rad) directly, your calculator needs to be in radian mode (not degree mode). ■ Use a sketch-graph to check that the answer is sensible. l fixed support m x x t 0 ε = 0 : x = A cos ( w t ) x t 0 ε = − −: x = A sin ( w t ) p 2 (a) (b) Fig. 3.2.5 Useful cases of x = A cos ( w t + e ) x / m 0.03 –0.03 0.2 0.4 0.6 0.8 t / s 0.55 0 0 Fig. 3.2.6 Graph for Example 1 Pointer The value of the constant, w , is fixed by the system itself (for a mass-spring system by k and m ). But A and e are determined by how the system is set in motion and timed. Pointer It’s (mercifully) rare to need any other version of x = A cos ( w t + e ) than x = A cos ( w t ) or x = A sin ( w t ) . Grade boost Don’t be afraid to draw x − t sketch-graphs. Sometimes they’re essential. d A metal ball of mass 0.12kg attached to a fixed point by a spring performs 43 cycles of oscillation in a minute . Calculate the spring constant. QUICKFIR QUICKFIRE QUICKFIRE 15 3.2 Vibrations