Question

A block of mass $M$ is placed on a smooth horizontal surface and it is pulled by a light spring as shown in the diagram. If the ends A and B of the spring are moving with $4\,ms^{- 1}$ and $2\,ms^{- 1}$ respectively and the rate at which spring energy is increasing is $20\,Js^{- 1}$ . then what is the value of spring force (in $N$ )?

Answer 1

Let, distance moved by $A$ and $B$ in any time interval as measured from natural length of spring are $x_A$ and $x_B$ respectively. Let, $ l $ is the natural length of spring, then elongation in spring at any instant is
$x =\left( x _{ A }- x _{ B }-l\right)$
$\text{U} = \frac{\text{kx}^{2}}{2}$
$\frac{\text{dU}}{\text{dt}} = \text{P} = \frac{\text{k}}{2} \times 2 \text{x} \frac{\text{dx}}{\text{dt}}$
$\Rightarrow 2 0 = \text{kx } \left[\text{v}_{\text{A}} - \text{v}_{\text{B}}\right]$
$\Rightarrow \text{kx} = \frac{2 0}{4 - 2} = 1 0\, \text{N}$