Why do I see some sources define the work on a spring as W = (1/2)(k)(xf^2 - xi^2) and some sources define it as W = (1/2)(k)(x^2) ?

Question

For example (I'll test out both equations):

Let xi = 3, xf = 7, and k = 3
So by using W = (1/2)(k)(xf^2 - xi^2)
                      = (1/2)(3)(7^2 - 3^2)
                      = (1/2)(3)(49 - 9)
                      = (1/2)(3)(40)
                      = 60

By using W = (1/2)(k)(x^2)
                  = (1/2)(k)(xf - xi)^2
                  = (1/2)(3)(7-3)^2
                  = (1/2)(3)(4)^2
                  = (1/2)(3)(16)
                  = 24

I'm confused since I didn't get the same result for both equations, so they are not the same then? Am I miscalculating something?

bobpursley · Accepted Answer

In the second equation, it assumes the initial x is zero. Work= INtegral force(x)*dx= INT k*x*dx over limits or work= 1/2 k (xf^2-xi^2)

Kid · Answer

Oh I think I get it now. So if xi = 0, then both versions of the equation will give the same result. Otherwise, if the initial position is not zero, then W = (1/2)(k)(xf^2 - xi^2) MUST be used. Correct?

bobpursley · Answer

correct. The second equation is for beginners (xi=0)