Question

A body of mass $m$ hangs by as inextensible light string that passes over a smooth massless pulley that is fitted with a light spring of stiffness $k$ as shown in the figure. If the body is released from rest, calculate the maximum elongation of the spring. Take $g=10{\text{m/s}}^2$.

• A. $\frac{mg}{k}$
• B. $\frac{2mg}{k}$
• C. $\frac{4mg}{k}$
• D. $\frac{3mg}{k}$

Answer 1

The correct option is C $\frac{4mg}{k}$


Let the maximum extension in the spring be $x$, then pulley will also fall through distance $x$
Applying string constraint:
$l_1+l_2=\text{constant}$
$\because$ string $l_1$ is fixed at one end, displacement at that end $=0$
Let mass $m$ travels by $x^{\prime }$ downwards
$\Rightarrow l_1+l_2=(l_1-x)+\left(\right(l_2-x)+x^{\prime })$
$\therefore x^{\prime }=2x$ (i)

Maximum elongation in spring will result when mass $m$ stops, i.e its velocity $v_f=0$
Applying energy conservation for the system:
Loss in gravitational PE of mass = Gain in elastic PE of spring
i.e $mg{x}^{\prime }=\frac{1}{2}k{x}^2$
$\Rightarrow mg\left(2x\right)=\frac{1}{2}k{x}^2$ (from (i))
$\therefore x=\frac{4mg}{k}$