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

It is given that:
Mass of disc = m
Radius of disc = r
The time period of torsional oscillations is T.
Moment of inertia of the disc at the centre, I$=\frac{m{r}^2}{2}$

Time period of torsional pendulum$\left(T\right)$ is given by,
$T=2\mathrm{\pi }\sqrt{\frac{I}{k}}$
where I is the moment of inertia, and
k is the torsional constant.

On substituting the value of moment of inertia in the expression for time period T, we have:
$$\begin{gathered} T=2\mathrm{\pi }\sqrt{\frac{m{r}^2}{2k}} \\ \mathrm{On}\mathrm{squaring}\mathrm{both}\mathrm{the}\mathrm{sides},\mathrm{we}\mathrm{get}: \\ {T}^2=4{\mathrm{\pi }}^2\frac{m{r}^2}{2k}=2{\mathrm{\pi }}^2\frac{m{r}^2}{k} \\ \Rightarrow 2{\mathrm{\pi }}^{2}m{r}^2=k{T}^2 \\ \Rightarrow k=\frac{2{\mathrm{\pi }}^{2}m{r}^2}{{\mathrm{T}}^2} \end{gathered}$$