Question

A highly rigid cubical block $A$ of small mass $M$ and side $L$ is fixed rigidly on to another cubical block $B$ of the same dimensions and of low modulus of rigidity $\eta$ such that the lower face of $A$ completely covers the upper face of $B$ . The lower face of $B$ is rigidly held on a horizontal surface. A small force $F$ is applied perpendicular to one of the side faces of $A$ . After the force is withdrawn, block $A$ executes small oscillations, the time period of which is given by

• A. $2 \pi \sqrt{M \eta L}$
• B. $2 \pi \sqrt{\frac{M \eta}{L}}$
• C. $2 \pi \sqrt{\frac{M L}{\eta}}$
• D. $2 \pi \sqrt{\frac{M}{\eta L}}$

Answer 1

Answer: (D)

Modulus of rigidity, $\eta = \frac{F}{A . \, \theta }$ Here, $A = L^{2}$
And
$\theta = \frac{\text{x}}{L}$
Therefore, restoring force is $F = - \eta \, A \theta = - \eta L x$

Or acceleration, $a=\frac{F}{M}=-\frac{\eta L}{M} x$
Since, $a \propto x$, oscillations are simaple harmonic in nature, time period of which is given by
$T=2 \pi \sqrt{\left|\frac{\text { displacement }}{\text { acceleration }}\right|}=2 \pi \sqrt{\left|\frac{x}{a}\right|}=2 \pi \sqrt{\frac{M}{\eta L}}$