Question

A highly rigid cubical block A of small mass M and side L is fixed rigidly onto another cubical block B of same dimensions and of low modulus of rigidly $\eta$ such that 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 A small force is applied perpendicular to one the side face of A. After the force is withdrawn, block A executes small oscillations, the time period of which is given by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. $\sqrt{\frac{M\eta }{L}}$

Answer 1

The correct option is B $2\pi \sqrt{\frac{M}{\eta L}}$

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

Here, $A=L^2$ and $\theta =\frac{x}{L}$

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 simple harmonic in nature, time period of which is given by,

$T=2\pi \sqrt{|\frac{Displacement}{Acceleration}|}=2\pi \sqrt{|\frac{x}{a}|}$

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