Question

Two blocks A and B of masses $m$ & $2m$ placed on smooth horizontal surface are connect with a light spring. The two blocks are given velocities as shown when spring is at natural length. Find maximum extension in the spring.

• A. $\sqrt{\frac{2m}{k}}v$
• B. $\sqrt{\frac{4m}{k}}v$
• C. $\sqrt{\frac{6m}{k}}v$
• D. $\sqrt{\frac{m}{k}}v$

Answer 1

The correct option is C $\sqrt{\frac{6m}{k}}v$

Given that,
$m_1=m$
$m_2=2m$
Reduced mass of the system.
$\mu =\frac{m_1{m}_2}{m_1+m_2}$
$\Rightarrow \mu =\frac{m\times 2m}{m+2m}$
$\Rightarrow \mu =\frac{2m}{3}$

External force on the system is zero.
So energy of the system is conserved.
Let maximum extension in the string is $x$.
$\frac{1}{2}\mu v_r^2=\frac{1}{2}k{x}^2$
$\Rightarrow \frac{1}{2}\times \frac{2m}{3}(3v{)}^2=\frac{1}{2}k{x}^2$
$\Rightarrow x=\sqrt{\frac{6m}{k}}v$