Question

Two blocks $A$ and B of masses $m$ and $2m$ respectively, attached at opposite ends of a spring of constant $K$ , placed on a smooth horizontal surface. Spring is initially at its natural length $l$ . A is given a velocity $2V_{0}$ and B given velocity $V_{0}$ as shown.



Maximum separation between $m$ and centre of mass of the system will be:

• A. $\frac{ l}{3}+\sqrt{\frac{8 m V_{0}^{2}}{3 K}}$
• B. $\frac{ l}{3}+\sqrt{\frac{2 m V_{0}^{2}}{3 K}}$
• C. $\frac{2 l}{3}+\sqrt{\frac{2 m V_{0}^{2}}{3 K}}$
• D. $\frac{2 l}{3}+\sqrt{\frac{8 m V_{0}^{2}}{3 K}}$

Answer 1

Answer: (D)

$l_{1}=\frac{2 m l}{2 m + m}=\frac{2 l}{3}$
$K_{1}\left(\textit{l}\right)_{1}=Kl\Rightarrow K_{1}\left(\frac{2 l}{3}\right)=Kl\Rightarrow K_{1}=\frac{3 K}{2}$
$\left|x_{1 \, m a x}\right|=\sqrt{\frac{m \left(2 V_{0}\right)^{2}}{\frac{3 K}{2}}}=\sqrt{\frac{8 m V_{0}^{2}}{3 K}}$
$\left|d_{1 \, m a x}\right|=l_{1}+x_{1 \, m a x}=\frac{2 l}{3}+\sqrt{\frac{8 m V_{0}^{2}}{3 K}}$