Question

A uniform rod of mass $m$ and length $l$ is suspended through a light wire of length $l$ and torsional constant $k$ as shown in figure. Find the time period if the system makes small oscillations in the vertical plane about the suspension point.

• A. $2\pi \sqrt{\frac{13l}{12g}}$
• B. $2\pi \sqrt{\frac{3l}{7g}}$
• C. $2\pi \sqrt{\frac{3l}{2g}}$
• D. $2\pi \sqrt{\frac{13l}{2g}}$

Answer 1

Answer: (A)

$2\pi \sqrt{\frac{13l}{12g}}$

The system makes small oscillations in vertical plane about suspension point which is also perpendicular to the plane of paper.
Moment of inertia of rod about this line by parallel axis theorem.
$I=\frac{m{l}^2}{12}+m{l}^2=\frac{13m{l}^2}{12}$

Time period, $T=2\pi \sqrt{\frac{I}{mgl}}=2\pi \sqrt{\frac{13m{l}^2}{12mgl}}=2\pi \sqrt{\frac{13l}{12g}}$