Question

When a rubber band is stretched by a distance $x$, it exerts a restoring force of magnitude $F=ax+b{x}^2$ where a and b are constants. The work done in stretching the un-stretched rubber band by $L$ is?

Answer 1

Restoring force:

1. Restoring force is defined as the force or torque that tends to restore a system or parts thereof to equilibrium after displacement.
2. The mathematical expression for restoring force is $F=-kx$, where $F$ is the restoring force, $k$ is the spring constant, $x$ is the displacement from equilibrium.
3. Work is defined as the force applied over a distance.
4. The mathematical expression for work is $W=F.sCos\theta$, where, $W$ is the work, $F$ is the force, $s$ is the distance, and$\theta$ is the angle between force and distance.
5. We have given that, the rubber band is stretched by a distance $x$ and restoring force is $F=ax+b{x}^2$.

So, the work done is :

$\mathrm{dW}=\mathrm{Fdx}$

${\int }_0^{\mathrm{W}}\mathrm{dW}={\int }_0^{\mathrm{L}}(\mathrm{ax}+{\mathrm{bx}}^2)\mathrm{dx}$

$W={\int }_0^L(ax+b{x}^2)dx$

$W=\frac{a{L}^2}{2}+\frac{b{L}^3}{3}$

Hence, the work done will be $\frac{a{L}^2}{2}+\frac{b{L}^3}{3}$.