How Is Work Calculated in a Stretched Spring?

[MrLiou168]

Homework Statement

A spring with stiffness k and unstretched length L is stretched so the elongation is d = x2 - L. A force is applied to make the final length of the spring x2. What is the work done by the force in terms of d?

Homework Equations

W = F * d = F*dx
d = x2 - L
F = k*dx

The Attempt at a Solution

Assuming W = F*dx and F = k*dx, then I derived F = k(x2 - L) = k*d

And plugging F back into the work equation, I got W = (kd)*d which is W = kd^2.

However, isn't the actual equation for work done by a spring W = (kx^2)/2? I can't seem to find where I missed the factor of 1/2. Any help greatly appreciated!

 

[Doc Al]

You assumed in your derivation that the force was constant and equal to its maximum value. Not so. As the spring is stretched, the force starts at zero and only reaches k*d at its full extension.

 

[MrLiou168]

Thanks Doc. So in this case would I simply integrate to find W? As in W = integral (F*dx)

and then W = integral(kxdx) = (kd^2)/2 ...?

 

[Doc Al]

Thanks Doc. So in this case would I simply integrate to find W? As in W = integral (F*dx)

and then W = integral(kxdx) = (kd^2)/2 ...?

Exactly.