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 η 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 After the force is withdrawn, block $A$ executes small oscillations.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 oscillation is

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

Answer 1

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

Let us try to solve this problem with the help of dimensional analysis first rather than going for the conventional approach.

Unit of $M=$ kg
Unit of η = $\frac{stress}{strain}=\frac{kgm{s}^{-2}}{m^2}=kg{m}^{-1}{s}^{-2}$
Unit of $L$ = m
So, on putting the units of these quantities in all the four options, we get option only (d) with the unit same as that of time (i.e $s$).