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. `aL^(2)+bL^(3)`
B. `(1)/(2)(aL^(2)+bL^(3))`
C. `(aL^(2))/(2)+(bL^(3))/(3)`
D. `(1)/(2)(aL^(2)/(2)+(bL^(3))/(3))`

Answer

C

Explanation

Work done in stretching the rubber-band by a distance dx is dW=Fdx =`(ax+bx^(2))dx`
Integrating both sides,
`W=underset(0)overset(L)intaxdx+underset(0)overset(L)intbx^(2)dx=(aL^(2))/(2)+(bL^(3))/(3)`