Question

Consider the situation shown in figure. A horizontal disc of mass m, radius R suspended from the centre through a wire of torsional rigidity k.If at $t=0$, the disc is deviated by ${\varphi }_0$, find the energy of small torsional vibrations:

• A. $\frac{1}{2}k{\varphi }_0^2$
• B. $\frac{1}{4}k{\varphi }_0^2$
• C. $k{\varphi }_0^2$
• D. $2k{\varphi }_0^2$

Answer 1

Answer: (A)

$\frac{1}{2}k{\varphi }_0^2$

Torque required to deviate the disc by an angle $\varphi$ $\tau =k\varphi$ where $k$ is the torsional rigidity

Now energy of torsional vibration for small angle $d\varphi$, $dE=\tau d\varphi$

$dE=k\varphi d\varphi$

Total energy of torsional vibration upto an angle ${\varphi }_o$, ${\int }_0^{E}dE={\int }_0^{{\varphi }_o}k\varphi d\varphi$

$E=k{\int }_0^{{\varphi }_o}\varphi d\varphi$

$E=k\times \frac{{\varphi }^2}{2}{|}_0^{{\varphi }_o}$

$E=k\frac{{\varphi }_o^2}{2}$