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) ?
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?
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)
correct. The second equation is for beginners (xi=0)
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?
You're correct that the two equations for the work done on a spring seem to give different results. This can be confusing, but let me explain why this discrepancy occurs.
The work done on a spring can be calculated using two different approaches: using displacement or using compression/extension.
The equation W = (1/2)(k)(xf^2 - xi^2) is derived using displacement. In this equation, xi represents the initial position of the spring, xf represents the final position of the spring, and k is the spring constant. The term (xf^2 - xi^2) represents the net change in displacement of the spring. By multiplying this net change in displacement by the spring constant divided by 2, you get the work done on the spring.
On the other hand, the equation W = (1/2)(k)(x^2), or W = (1/2)(k)(xf - xi)^2, is derived using compression/extension. In this equation, x represents the compression or extension of the spring, which is simply the difference between its final and initial positions. This equation is obtained by squaring the compression/extension and multiplying it by the spring constant divided by 2.
Now, let's plug in the values you provided and calculate the work using both approaches:
Using the equation W = (1/2)(k)(xf^2 - xi^2):
W = (1/2)(3)(7^2 - 3^2)
= (1/2)(3)(49 - 9)
= (1/2)(3)(40)
= 60
Using the equation W = (1/2)(k)(x^2):
W = (1/2)(3)(7 - 3)^2
= (1/2)(3)(4)^2
= (1/2)(3)(16)
= 24
As you can see, we obtained different results using the two equations. This is because the equations are derived based on different approaches (displacement vs. compression/extension).
So, it's important to use the appropriate equation depending on the given information. If you're given the initial and final positions of the spring, you can use the displacement equation (W = (1/2)(k)(xf^2 - xi^2)). But if you're given the compression or extension of the spring, you should use the compression/extension equation (W = (1/2)(k)(x^2)).
Remember, it's not that either equation is incorrect; they simply represent different ways of calculating the work done on a spring based on the available information.