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Question

Physics Question on work, energy and power

When a rubber -band is stretched by a distance x, it exerts a restoring force of magnitude F=ax+bx2 F = ax + bx^2, where a and b are constants. The work done in stretching the unstretched rubber-band by LL is :

A

aL2+bL3aL^2 + b L^3

B

12(aL2+bL3) \frac{1}{2}(aL^2 + b L^3)

C

aL22+bL33\frac{a L^2}{2} + \frac{ bL^3}{ 3}

D

12(aL22+bL33)\frac{1}{2}\left(\frac{a L^{2}}{2}+\frac{b L^{3}}{3}\right)

Answer

aL22+bL33\frac{a L^2}{2} + \frac{ bL^3}{ 3}

Explanation

Solution

dW=Fdl\int d W=\int F \cdot d l
W=0Laxdx+0Lbx2dxW=\int_{0}^{L} a x d x+\int_{0}^{L} b x^{2}\, dx
=aL22+bL33=\frac{a L^{2}}{2}+\frac{b L^{3}}{3}