Question

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

• A. $aL^2 + b L^3$
• B. $ \frac{1}{2}(aL^2 + b L^3)$
• C. $\frac{a L^2}{2} + \frac{ bL^3}{ 3}$
• 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
$U_{f}-U_{i}=-W=-\int\limits_{t}^{f} F \cdot d r$
Given, $F=a x+b x^{2}$
We know that work done in stretching the rubber band by $L$ is $|d W|=|F d x|$
$|W| =\int\limits_{0}^{L}\left(a x+b x^{2}\right) d x$
$=\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] $
$=|W|=\frac{a L^{2}}{2}+\frac{b L^{3}}{3}$