Question

Two torsion pendula (P, Q) are shown. They differ only by the addition of cylindrical masses. The radius of each additional mass is $\frac{1}{4}$th of the radius of the disc. If each cylinder and disc have equal mass, then

• A. moment of inertia of Q is $\frac{16}{27}$MR${}^2$
• B. moment of inertia of Q is $\frac{27}{16}$MR${}^2$
• C. ratio of periods of P and Q is ${\left(\frac{3}{4}\right)}^{\frac{1}{2}}$
• D. ratio of periods of P and Q is ${\left(\frac{2}{3}\right)}^{\frac{3}{2}}$

Answer 1

Answer: (B), (D)

moment of inertia of Q is $\frac{27}{16}$MR${}^2$
ratio of periods of P and Q is ${\left(\frac{2}{3}\right)}^{\frac{3}{2}}$

$I_1\theta +A\theta =0$
$I_2\theta +A\theta =0$
where A is restoring coefficient of each time
${\omega }_1=\sqrt{\frac{A}{I_1}}$
${\omega }_2=\sqrt{\frac{A}{I_2}}$
$I_1=\frac{M{R}^2}{2}$
$I_1=\frac{M{R}^2}{2}+2[\frac{M}{2}{\left(\frac{R}{4}\right)}^2+M{\left(\frac{3R}{4}\right)}^2]=\frac{27}{16}M{R}^2$
Ratio of periods,
$\frac{T_1}{T_2}=\frac{{\omega }^2}{{\omega }^1}=\sqrt{\frac{I_1}{I_2}}$
$\Rightarrow \frac{T_1}{T_2}={\left(\frac{2}{3}\right)}^{\frac{3}{2}}$