Question

A solid disk of radius $R$ is suspended from a spring of linear spring constant $k$ and torsional constant $C$, as shown in figure. In terms of $k$ and $C$, what value of $R$ will give the same period for the vertical and torsional oscillations of this system?

• A. $\sqrt{\frac{2C}{k}}$
• B. $\sqrt{\frac{C}{2k}}$
• C. $2\sqrt{\frac{C}{k}}$
• D. $\frac{1}{2}\sqrt{\frac{C}{k}}$

Answer 1

The correct option is A $\sqrt{\frac{2C}{k}}$


Time - period $\left(T\right)$ for linear SHM
$T=2\pi \sqrt{\frac{m}{k}}...\left(i\right)$


Time - period $\left(T\right)$ for angular SHM
$T=2\pi \sqrt{\frac{I}{C}}...\left(ii\right)$
For disc

$I_C=\frac{m{R}^2}{2}$
From (i) and (ii)
$2\pi \sqrt{\frac{m}{k}}=2\pi \sqrt{\frac{m{R}^2}{2C}}$
$\frac{m}{k}=\frac{m{R}^2}{2C}$
$R=\sqrt{\frac{2C}{k}}$
Hence, option (A) is correct.