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 $4m{s}^{-1}$ and $2m{s}^{-1}$ respectively and the rate at which spring energy is increasing is $20J{s}^{-1}$ . then what is the value of spring force (in $N$ )?

Answer 1

Answer: 10

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
$x=\left(x_A-x_B-l\right)$
$\text{U}=\frac{{\text{kx}}^2}{2}$
$\frac{\text{dU}}{\text{dt}}=\text{P}=\frac{\text{k}}{2}\times 2\text{x}\frac{\text{dx}}{\text{dt}}$
$\Rightarrow 20=\text{kx }\left[{\text{v}}_{\text{A}}-{\text{v}}_{\text{B}}\right]$
$\Rightarrow \text{kx}=\frac{20}{4-2}=10\text{N}$