Question

An insect of negligible mass is sitting on a block of mass m, tied to an ideal spring as shown in the figure. The force constant of spring is k. The block performs simple harmonic motion with amplitude A in front of a plane mirror placed as shown in the figure. The maximum speed of insect relative to its image will be

• A. $A\sqrt{\frac{k}{m}}$
• B. $\frac{A}{\sqrt{2}}\sqrt{\frac{k}{m}}$
• C. $A\sqrt{2}\sqrt{\frac{k}{m}}$
• D. $A\sqrt{\frac{m}{k}}$

Answer 1

The correct option is C $A\sqrt{2}\sqrt{\frac{k}{m}}$


Velocity of object $=v_{max}\frac{1}{\sqrt{2}}\hat{i}+V_{max}\frac{1}{\sqrt{2}}\hat{j}$
Velocity of image $=V_{max}\frac{1}{\sqrt{2}}\hat{i}-V_{max}\frac{1}{\sqrt{2}}\hat{j}$
${\overline{V}}_r={\overline{V}}_{object,image}=(V_{max}\frac{1}{\sqrt{2}}\hat{i}+V_{max}\frac{1}{\sqrt{2}}\hat{j})-(V_{max}\frac{1}{\sqrt{2}}\hat{i}-V_{max}\frac{1}{\sqrt{2}}\hat{j})$
$=V_{max}\sqrt{2}\hat{j}$
$|{\overline{V}}_r|=\sqrt{2}{V}_{max}=\sqrt{2}A\omega (\because V_{max}=A\omega )$
$=\sqrt{2}A\sqrt{\frac{k}{m}}(\because \omega =\sqrt{\frac{k}{m}})$