Question

Consider the situation shown in figure. Initially, the spring is unstretched when the mass $m$ is released from rest. Assuming no friction in the pulley, find the maximum extension in the spring.

• A. $\frac{mg}{k}$
• B. $\frac{mg}{2k}$
• C. $\frac{2mg}{k}$
• D. $\frac{4mg}{k}$

Answer 1

The correct option is A $\frac{mg}{k}$


At the maximum extension, mass $m$ must finally come to rest.
Therefore, $\Delta K.E=0$

$W_{ext}=\Delta K.E+\Delta U$
There is no external force on the system
So, for maximum extension in the spring, $\Delta U=0$
Loss in gravitational $P.E$ of mass = Gain in elastic $P.E$ energy of spring

From string constraint on pulley,
extension of spring = $2\times$ displacement of mass
$\Delta U=0$
$\Rightarrow \frac{1}{2}k{x}^2=mg\frac{x}{2}$
$\Rightarrow x=\frac{mg}{k}$