First slide

Work

Question

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

Moderate

A

$\frac{1}{2} (\frac{a L^{2}}{2} + \frac{b L^{3}}{3})$
B

$a L^{2} + b L^{3}$
C

$\frac{1}{2} (a L^{2} + b L^{3})$
D

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

Solution

$G i v e n r e s t o r i n g F o r c e o f R u b b e r B a n d i s F = a x + b x^{2} - - - (1) w o r k d o n e i n s t r e t c h i n g t h e R u b b e r B a n d i s d w = F . d x l e n g t h e x t e n d e d f r o m 0 t o L ∴ w o r k d o n e = W = \int_{0}^{L} d w W = \int_{0}^{L} F . d x s u b s t i t u t e e q n (1) W = \int_{0}^{L} (a x + b x^{2}) . d x W = a {(\frac{x^{2}}{2})}_{0}^{L} + b {(\frac{x^{3}}{3})}_{0}^{L} ∴ W = \frac{a L^{2}}{2} + \frac{b L^{3}}{3} = w o r k d o n e i n s t r e t c h i n g R u b b e r B a n d f r o m 0 t o L$

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