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 After the force is withdrawn, block $A$ executes small oscillations, the time period of which is given by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. $2\pi \sqrt{\frac{ML}{\eta }}$
• 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 $F=-\eta A\theta =-\eta Lx$

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
$T=2\pi \sqrt{\left|\frac{\text{ displacement }}{\text{ acceleration }}\right|}=2\pi \sqrt{\left|\frac{x}{a}\right|}=2\pi \sqrt{\frac{M}{\eta L}}$