# Homework Help: Work of a Spring Derivation

1. Jan 29, 2013

### MrLiou168

1. The problem statement, all variables and given/known data
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?

2. Relevant equations
W = F * d = F*dx
d = x2 - L
F = k*dx

3. 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!

2. Jan 29, 2013

### Staff: Mentor

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.

3. Jan 29, 2013

### 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 ...?

4. Jan 29, 2013

Exactly.