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. \[\dfrac{{a{L^2}}}{2} + \dfrac{{b{L^3}}}{3}\]
B. \[\dfrac{1}{2}\left( {\dfrac{{a{L^2}}}{2} + \dfrac{{b{L^3}}}{3}} \right)\]
C. \[a{L^2} + b{L^3}\]
D. \[\dfrac{1}{2}\left( {a{L^2} + b{L^3}} \right)\]

Answer 1

Hint:The above problem can be resolved using the restoring forces' concept and fundamentals and the work done against the restoring forces. The rubber band is stretched to some distance for a given problem; some magnitude of the force is applied. This magnitude of the force is known as applied force, and the same amount of force is restored and is known as restoring force. The mathematical relation for the restoring force is integrated to calculate the work against the restoring force.

Complete step by step answer:
The expression for the work done in stretching the unstretched rubber band by L is given as,
\[dW = Fdx\]
Integrating the above equation as,
 \[
\int\limits_0^W {dW} = \int\limits_0^L {\left( {ax + b{x^2}} \right)dx} \\
\Rightarrow W = a\left[ {\dfrac{{{x^2}}}{2}} \right]_0^L + b\left[ {\dfrac{{{x^3}}}{3}} \right]_0^L\\
\therefore W = a\dfrac{{{L^2}}}{2} + b\dfrac{{{L^3}}}{3}
\]
Therefore, the work done in stretching is \[\dfrac{{a{L^2}}}{2} + \dfrac{{b{L^3}}}{3}\] and option (A) is correct.

Note: Try to understand the restoring force's concept and meaning, and the work is done against the restoring force. The practical applications of the restoring forces can be observed when there is some stretching activity is considered. Some energy is stored within the band during stretching activities, and this energy is known as the elastic potential energy. Moreover, there is wide application of the concepts of restoring force, like in case of the spring and many more. The spring force is that force which is constituted by external force, and his spring force is related with the material stiffness commonly known as the spring stiffness. Furthermore, the restoring force is applied to various industrial applications also.