Question

A simple harmonic oscillator has an amplitude of 0.17 m and a period of 0.84 s.
Determine the frequency, and the angular frequency of the motion. Write expressions for
the time dependence of displacement, velocity and acceleration. What are the maximum
values of the velocity and acceleration?

Answer 1

Question 21562

The amplitude $A = 0.17m$ , period $T = 0.84s$ .

By definition, angular frequency $\omega = \frac{2\pi}{T} = 7.48\frac{1}{s}$ , and frequency is equal to

$$v = \frac {1}{T} = 1.19 \frac {1}{s}.$$

The expression for displacement is a sine or cosine wave, with given parameters. Let us choose cosine function:

$$x (t) = A \cos (\omega t + \delta) = 0.17 \cos (7.48 t + \delta).$$

Velocity and acceleration are first and second derivatives of displacement respectively.

Hence,

Velocity: $v = \dot{x} = -A\omega \sin (\omega t + \delta) = -1.27\sin (7.48t + \delta)$ , and

Acceleration: $a = \ddot{x} = -A\omega^2\cos (\omega t + \delta) = 9.51\cos (7.48t + \delta)$ .