Question

Two rings of mass $m$ and $2m$ are connected with a massless spring and can slip over two frictionless parallel horizontal rails having separation equal to the unstretched length of the spring as shown in figure. Ring of mass $m$ is given velocity $v_{0}$ in the direction shown. Maximum length by which the spring will be stretched is

• A. $\sqrt{\frac{m}{k}}v_{0}$
• B. $\sqrt{\frac{3 m}{k}}v_{0}$
• C. $\sqrt{\frac{2 m}{3 k}}v_{0}$
• D. $\sqrt{\frac{2 m}{k}}v_{0}$

Answer 1

Answer: (C)

Maximum expansion takes place only when both the rings move with the same speed.
Therefore we can apply work-energy theorem here. So maximum expansion in spring is obtained by,
$\frac{1}{2}kx_{m a x}^{2}=\frac{1}{2}\mu v_{0}^{2} \, \, \left[\mu = R e d u c e d \, m a s s\right]$
$\Rightarrow \, \, \, x_{m a x}=\sqrt{\frac{\mu }{k}}.v_{0}=\sqrt{\frac{2 m}{3 k}}v_{0}$