Question

Find an expression for the torsional constant k in terms of the moment of inertia I of the disk and the angular frequency ω of small, free oscillations.

Answer 1

Answer:

Explanation:

The general equation for the disk with moment of inertia I when given small angular displacement  $\theta$ is given by

$I\frac{\mathrm{d^2} \theta }{\mathrm{d} t^2}=-k\theta$

$\frac{\mathrm{d^2} \theta }{\mathrm{d} t^2}+\frac{k\theta }{I}=0$

Replacing

$\frac{k\theta }{I}=\omega ^2$

where $\omega$ is the angular frequency of oscillation

General solution for this Equation is given by

$\theta =\theta _{max}\sin \left ( \omega t+\phi \right )$

where $\theta _{max}=maximum\ angular\ displacement$

$\phi =Phase\ difference$

Thus K can be written as

$k=I\omega ^2$