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. $\frac{a{L}^2}{2}+\frac{{bL}^3}{3}$
• B. $\frac{1}{2}(\frac{a{L}^2}{2}+\frac{{bL}^3}{2})$
• C. $a{L}^2+{bL}^3$
• D. $\frac{1}{2}(a{L}^2+{bL}^3)$

Answer 1

The correct option is A $\frac{a{L}^2}{2}+\frac{{bL}^3}{3}$

Work done by streching dx length is,

$dw=Fdx$

$=(ax+b{x}^2)dx$

$w={\int }_0^L(ax+b{x}^2)dx=\stackrel{L}{\underset{0}{[\frac{{ax}^2}{2}+\frac{{bx}^3}{3}]}}=\frac{{aL}^2}{2}+\frac{{bL}^3}{3}$