Question

A uniform stick of length $l$ is mounted so as to rotate about a horizontal axis perpendicular to the stick and at a distance $d$ from the centre of mass. The time period of small oscillations has a minimum value when $\frac{d}{l}$ is:

• A. $\frac{1}{\sqrt{2}}g$
• B. $\frac{1}{\sqrt{12g}}$
• C. $\frac{1}{\sqrt{3}}$
• D. $\frac{1}{\sqrt{6}g}$

Answer 1

Answer: (B)

$\frac{1}{\sqrt{12g}}$

$T=2\pi \sqrt{\frac{I}{mgd}}$
$T=2\pi \sqrt{\frac{\frac{m{l}^2}{12}+m{d}^2}{mgd}}$
$T=2\pi \sqrt{\frac{l^2+12d^2}{12gd}}$
$T=2\pi \sqrt{(\frac{l^2}{12gd}+d)}$
$T$ is minimum when fIrst derivative of the quantity inside the root with respect to $d$ is zero.
$-\frac{l^2}{12g{d}^2}+1=0$
or $\frac{d}{l}=\frac{1}{\sqrt{12g}}$