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.

 

[rezeew]

Your derivation is correct in terms of the work done by the force on the spring. However, the equation for work done by a spring is actually W = (kx^2)/2, as you mentioned. This is because the force applied to the spring is not constant as it stretches, but rather it increases as the spring stretches due to Hooke's Law (F = kx). Therefore, the work done must take into account the varying force and is given by the integral of F*dx, which results in the (kx^2)/2 term. So, your initial derivation is correct, but the final equation for work done by the force should be W = (kx^2)/2.