In simple harmonic motion and .

PHYSICS · Multi-part · A Level · Physics · Further mechanics and thermal physics · Further mechanics and thermal physics

Problem

In simple harmonic motion

x = x_0 \sin(\omega t + \phi)

and

v = \omega x_0 \cos(\omega t + \phi)

. (a) Using the identity

\sin^2 \theta + \cos^2 \theta = 1

for

\theta = \omega t + \phi

, show that

v^2 = \omega^2 (x_0^2 - x^2)

. (b) The maximum speed in a simple harmonic oscillation is

v

. What is the speed when the displacement is half the amplitude?