A highly rigid cubical block $A$ of small mass $M$ and side $L$ is fixed rigidy on the othe-cubical block of…

A highly rigid cubical block $A$ of small mass $M$ and side $L$ is fixed rigidy on the othe-cubical block of 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 After the force is withdrawn, block $A$ executes small oscillations, the time period of which is given by
  1. $2 \pi \sqrt{M \pi L}$
  2. $\quad 2 \pi \sqrt{(M n / L)}$
  3. $2 \pi \sqrt{M L / n}$
  4. $\quad 2 \pi \sqrt{M / \eta L}$

Solution

Correct option is(d) $2 \pi \sqrt{\mathrm{M} / \eta \mathrm{L}}$
Modulus of rigidity $\eta=\frac{\mathrm{F}}{\mathrm{A} \theta}$
Here, $\mathrm{A}=\mathrm{L}^{2}$
and $\theta=\frac{\mathrm{x}}{\mathrm{L}}$ for small $\theta$
$\therefore$ Restoring force $=\mathrm{F}=-\eta \mathrm{A} \theta$
or acceleration, $a=\frac{\mathrm{F}}{\mathrm{m}}=\frac{\eta \mathrm{L}}{\mathrm{M}} \mathrm{x} \quad$ equation $(1)$
$\because \mathrm{a} \propto(-\mathrm{x})$
$\therefore$ Time period, $\mathrm{T}=2 \pi \sqrt{\left|\frac{\mathrm{x}}{\mathrm{a}}\right|}$
$\Longrightarrow \mathrm{T}=2 \pi \sqrt{\frac{\mathrm{M}}{\mathrm{\eta} \mathrm{L}}}$ /

Asked in: JEE Mains - Units and Dimensions - Test 2

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