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 A small force F is applied perpendicular to one of a sides faces of A. After the force is withdrawn, block A executes small oscillations, the time period of which is given by.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. $2\pi \sqrt{M\eta /L}$
• C. $2\pi \sqrt{ML/\eta }$
• D. $2\pi \sqrt{M/\eta L}$

Answer 1

The correct option is D $2\pi \sqrt{M/\eta L}$

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

Here, $A=L^2$

and $\theta =\frac{x}{L}$ for small $\theta$

$\therefore$ Restoring force $=F=-\eta A\theta$

or acceleration, $a=\frac{F}{m}=\frac{\eta L}{M}xequation\left(1\right)$

$\because a\propto (-x)$

$\therefore$ Time period, $T=2\pi \sqrt{\left|\frac{x}{a}\right|}$

$⟹T=2\pi \sqrt{\frac{M}{\eta L}}$