Question

A uniform disc of mass m and radius r is suspended through a wire attached to its centre. If the time period of the torsional oscillations be T, what is the torsional constant of the wire ?

Answer 1

M.I. of the centre of the disc = $\frac{m{r}^2}{2}$

$T=2\pi \sqrt{\frac{I}{k}}=2\pi \frac{m{r}^2}{2K}$

[where K = Torisional constant]

$T^2=4{\pi }^2\frac{m{r}^2}{2K}=2{\pi }^2\frac{m{r}^2}{K}$

$\Rightarrow 2{\pi }^{2}m{r}^2=KT$

$\Rightarrow K=\frac{2m{r}^2{\pi }^2}{T^2}$

$\therefore$ Torsional constant,

$K=\frac{2{\pi }^{2}m{r}^2}{T^2}$