Question

The bob of a simple pendulum, which is in the shape of a hollow cylinder of mass $M$, radius $r$ and length $h$ is suspended by a long string (the mass of the base and lid of the cylinder are negligible). The cylinder is filled with a liquid of density $\rho$ upto a height of $x$. Then the value of $x$ for which the time period of the pendulum is maximum, is given by which of the following equations: $\left(\lambda =\pi r^2\rho \right)$

• A. $\lambda x^2-2Mx+Mh=0$
• B. $\lambda x^2+2Mx+Mh=0$
• C. $x=H/2$
• D. $\lambda x^2+2Mx-Mh=0$

Answer 1

Answer: (D)

Centre of mass of system is
$x_{cm}=\frac{(\lambda x)\frac{x}{2}+M\left(\frac{h}{2}\right)}{M+\lambda x}$
Time period is maximum when
$\frac{d{x}_{cm}}{dx}=0$
$\Rightarrow (M+\lambda x)\lambda x-\left(\frac{\lambda x^2}{2}+\frac{Mh}{2}\right)\lambda =0$
$\left(Mx+\lambda x^2\right)-\frac{\lambda x^2}{2}-\frac{Mh}{2}=0$
$\lambda x^2+2Mx-Mh=0$