Question

A highly rigid cubical block $A$ of small mass $M$ and side $L$ is fixed rigidly on to another cubical block of same dimensions and of low modulus of rigidity $\eta$ such that the lower face of $A$ completely covers the upper face $B$. The lower face of $B$ is rigidly held on a horizontal surface. A small force $F$ is applied perpendicular to one of the side face 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 After the force is withdrawn, block $A$ executes small oscillations, the time period of which is given by :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. $2\pi \sqrt{\frac{M}{L\eta }}$

Answer 1

Answer: (D)

$2\pi \sqrt{\frac{M}{L\eta }}$

Modulus of rigidity $\eta =\frac{E}{A\theta }$

We know, $A=L^2$

For small $\theta$,

$\theta =\frac{x}{L}$

Restoring force $F=-\eta AB$

$F=-\eta Lx$

Acceleration, $a=-\frac{\eta L}{M}x\to \left(1\right)$

$a=\frac{F}{m}$

Here, $\alpha \propto -x$

$\therefore$here we have SHM,

Hence the time period is given by

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

where,

$a$=acceleration and $x$=small deformation.