Question

A uniform disc of radius $5.0\text{cm}$ and mass $200\text{g}$ is fixed at its centre to a metal rod, the other end of which is fixed to a celling. The hanging disc is rotated about the rod through an angle and is released. If the disc makes torsional oscillations with time period $0.20\text{s}$, find the torsional constant of the rod.

• A. $0.50{\text{kg m}}^2{\text{/s}}^2$
• B. $0.75{\text{kg m}}^2{\text{/s}}^2$
• C. $0.125{\text{kg m}}^2{\text{/s}}^2$
• D. $0.25{\text{kg m}}^2{\text{/s}}^2$

Answer 1

The correct option is D $0.25{\text{kg m}}^2{\text{/s}}^2$

Moment of inertia of the disc about an axis passing through the rod is
$I=\frac{m{r}^2}{2}=\frac{\left(0.200\text{kg}\right)(5.0\times {10}^{-2}{m}^2{)}^2}{2}$
$I=2.5\times {10}^{-4}{\text{kg - m}}^2$
The time period is given by
$T=2\pi \sqrt{\frac{I}{k}}$
$k=\frac{4{\pi }^{2}I}{T^2}$
$k=\frac{4{\pi }^2(2.5\times {10}^{-4}{\text{kg-m}}^2)}{(0.20s{)}^2}=0.25{\text{kg-m}}^2{\text{/s}}^2$