Work Energy And Power Question 146
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,
Options:
A) $ \frac{aL^{2}}{2}+\frac{bL^{3}}{3} $
B) $ \frac{1}{2}( \frac{aL^{2}}{2}+\frac{bL^{3}}{3} ) $
C) $ aL^{2}+bL^{3} $
D) $ \frac{1}{2}(aL^{2}+aL^{3}) $
Show Answer
Answer:
Correct Answer: A
Solution:
[a] Restoring force on rubber-band, $ F=ax+bx^{2} $
Work done in stretching the rubber-band by a small amount $ dx,dW=Fdx $
Net work done in stretching the rubber-band by L is
$ W=\int{dW=\int\limits_0^{L}{Fdx=\int\limits_0^{L}{(ax+bx^{2})dx}}} $
$ \Rightarrow W={{[ a\frac{x^{2}}{2}+b\frac{x^{3}}{3} ]}_0}^{L}=\frac{aL^{2}}{2}+\frac{bL^{3}}{3} $
