Question

(a)Show that the average kinetic energy associated with SHM is .1/4ka²
b) A rubber pad acts as an elastic spring. When a mass of 200 g is placed on it, a compression of 2.0 cm is produced. The spring-mass system begins to oscillate when the mass is gently tapped downwards. Calculate the frequency of oscillation.

Answer 1

a)

$$K(t)=\frac{1}{2}ma^2\omega^2\cos^2(\omega t)=\frac{1}{2}ka^2\cos^2(\omega t)$$

$$K_{av}=\frac{1}{2T}ka^2\int^T_0\cos^2(\omega t)dt$$

$$\int^T_0\cos^2(\omega t)dt=\frac{T}{2}$$

$$K_{av}=\frac{1}{4}ka^2$$

b)

$$kx=mg$$

$$k=\frac{mg}{x}$$

The frequency:

$$f=\frac{1}{2\pi}\sqrt{\frac{k}{m}}=\frac{1}{2\pi}\sqrt{\frac{mg}{mx}}=\frac{1}{2\pi}\sqrt{\frac{g}{x}}$$

$$f=\frac{1}{2\pi}\sqrt{\frac{9.8}{0.02}}=3.5\ Hz$$