WJEC Physics for A2: Study and Rev Guide

Example 2 A piston in an engine is moving with shm, of amplitude 0.040m , and frequency 35Hz . At one point in its cycle its upward displacement from its mid position is 0.024m , and is increasing. (a) Find how long it takes to reach this point from its mid position. (b) How much longer will it be before this displacement occurs again? Answer (a) Choose e = − p 2 , to make t = 0 when piston passes through its mid position, going up. (We’ll call upwards the + x direction.) Then x = A sin( w t ) , so sin( w t ) = x A = 0.024 m 0.040 m = 0.600 So w t = sin −1 (0.600) = 0.644 rad (calculator in radian mode!) So t = 0.644 w = 0.644 2 p f = 0.644 2 p 35 s −1 = 2.93 ms. To 2 sf, t = 2.9ms A quick check: T 4 = 1 4 × 1 f = 1 4 × 1 35 s −1 = 7.14ms ; 2.93ms< 7.14ms ; con rming that 2.93 s is during the rst upstroke. (b) The ‘inverse sine’ ( sin −1 ) function on a calculator gives only the angle (with the given sine) in the range − p 2 to + p 2 . To nd any more times, after 2.9ms , when x = 0.024m , you need to draw a circle diagram or – as here – a sketch-graph (Fig. 3.2.7), making use of its symmetry. The extra time elapsing is seen to be 8.42ms ( 8.4ms to 2 sf). Note the use of T 4 = 7.14ms . x / m 0.040 0.024 0 4.2 4.2 2.93 7.14 t / ms Fig. 3.2.7 Graph for Ex 2 (b) e f g h A metal ball hangs from a long thread, as a pendulum. The ball is pulled to one side from its rest position by 0.040m in the + x direction and released. It performs shm of period 2.00 s . QUICKFIRE QUICKFIRE QUICKFIRE e Calculate the pendulum length, l . QUICKFIRE QUICKFIRE QUICKFIRE f Calculate the displacement, x , of the ball 1.20 s after release. QUICKFIRE QUI IRE QUICKFIRE g Determine x at 1.50 s after release. [Easy by sketch-graph!] QUICKFIRE QUICKFIRE QUICKFIRE h Calculate the times from release at which x is −0.020 m : a) for the first time b) for the next time. QUICKFIRE QUICKFIRE QUICKFIRE 16 A Level Physics: Study and Revision Guide