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 unstretched rubber-band by $L$ is :

• A. $a{L}^2+b{L}^3$
• B. $\frac{1}{2}(a{L}^2+b{L}^3)$
• C. $\frac{a{L}^2}{2}+\frac{b{L}^3}{3}$
• D. $\frac{1}{2}\left(\frac{a{L}^2}{2}+\frac{b{L}^3}{3}\right)$

Answer 1

Answer: (C)

Thinking Process We know that change in potential energy of a system corresponding to a conservative internal force as
$U_f-U_i=-W=-\underset{t}{\overset{f}{\int }}F\cdot dr$
Given, $F=ax+b{x}^2$
We know that work done in stretching the rubber band by $L$ is $|dW|=|Fdx|$
$|W|=\underset{0}{\overset{L}{\int }}\left(ax+b{x}^2\right)dx$
$={\left[\frac{a{x}^2}{2}\right]}_0^L+{\left[\frac{b{x}^3}{3}\right]}_0^L$
$=\left[\frac{a{L}^2}{2}-\frac{a\times (0{)}^2}{2}\right]+\left[\frac{b\times L^3}{3}-\frac{b\times (0{)}^3}{3}\right]$
$=|W|=\frac{a{L}^2}{2}+\frac{b{L}^3}{3}$