Question

A block of mass 'm' is pushed against a spring of spring constant K fixed at other end to a wall. The block can slide of a frictionless table as shown. The nature length of the origin is $L_o$. It is compressed to half its natural length and leased. Find the velocity of the block as a function of its distance from the wall.

Answer 1

Given,

Initial natural length,= $L_o$

There initial compression = $L_0/2$
Energy stored in the spring, $=\frac{1}{2}K{\left(\frac{L_o}{2}\right)}^2=\frac{1}{8}K{L}_0^2$
This it moves small distance X, say the velocity it has is v and so
$\frac{1}{2}K{[\frac{L_0}{2}-X]}^2+\frac{1}{2}m{v}^2=\frac{1}{8}K{L}_0^2$
$\Rightarrow \frac{1}{2}K{X}^2-\frac{1}{2}K\left(L_{o}X\right)+\frac{1}{2}m{v}^2=0$
$\Rightarrow K{X}^2-K{L}_{o}X+m{v}^2=0$
${\text{v}}^2=\frac{[L_{o}X-X^2]K}{m}$
$\text{v}=\sqrt{\frac{K(L_{o}X-X^2)}{m}}$

Hence, velocity and distance relation is $\sqrt{\frac{K(L_{o}X-X^2)}{m}}$