# Where does 1/2 come from in 1/2kx^2?

Where does 1/2 come from in 1/2kx^2??

Sorry is this sounds like a silly question, but i was just wondering where the 1/2 in PE = 1/2kx^2 comes from. If F=kx, then wouldnt it be kx*x = kx^2 ?

arildno
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In this case the (force*distance) formula must be represented as an integral, for example:

$$PE=\int_{0}^{x}kx'dx'=\frac{1}{2}kx^{2}$$

errr....I havnt taken calculas yet, so i have no idea what that means, lol.

Doc Al
Mentor
force is not constant

ArmoSkater87 said:
errr....I havnt taken calculas yet, so i have no idea what that means, lol.
Even without calculus you should be able to see where the half comes in. In the equation Work = Force * Distance, the force is changing so you can't just put F = kx, since that's only true when the spring is fully stretched. Instead, use the average force: The force varies uniformly from 0 to kx, so the average force is kx/2. So Work = (kx/2)*(x) = 1/2 k x^2. Make sense?

basically, what arildno did was integrate. You dont need much calculus to understand. If you have
$$F = kx$$

now you find the anit - derivative, do you know what that means?

$$P_e = \frac {1}{2}kx^2$$

if you still dont understand, pm me.

chroot
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It doesn't take much calculus to perform an integration, Nenand?

- Warren

chroot said:
It doesn't take much calculus to perform an integration, Nenand?

- Warren

chill, chill, chroot, hmmm,..., Nenand, chroot ???

he was only saying integrating a constant multiplied by x is one of them elementary integranda...

marlon

robphy
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Plot force F(x) vs displacement x.
The "area under the curve" is the "work done by that force".

Since F(x) is linear, you are finding the area of a triangle with base (x) and height (-kx).
So, (Work done by F)=(1/2)(x)(-kx).

Since that force is conservative, the potential energy is minus the work done by that force.
So, U=-(-(1/2)kx^2)=(1/2)kx^2.

chroot said:
It doesn't take much calculus to perform an integration, Nenand?

- Warren
integration is calculus. And I wasnt shure if he had any experience with polynomial functions or calculus.

chroot
Staff Emeritus
Gold Member
integration is calculus.
I'm aware.
And I wasnt shure if he had any experience with polynomial functions or calculus.
He said he didn't.

- Warren

Oh wow, that makes sense, at least the explanations without the use of "intergration" or "derivative" and that calculus stuff. I have no idea what those words mean, although ive heard them many times. I'm going to take BC calc when school starts. Anyways, thanks a lot for the responses. :)

russ_watters
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ArmoSkater87 said:
Oh wow, that makes sense, at least the explanations without the use of "intergration" or "derivative" and that calculus stuff. I have no idea what those words mean, although ive heard them many times. I'm going to take BC calc when school starts. Anyways, thanks a lot for the responses. :)
To get you started, an integral is the area under a curve and the derivative is the slope of the curve. Its more complicated than that, of course...

NateTG
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ArmoSkater87 said:
Sorry is this sounds like a silly question, but i was just wondering where the 1/2 in PE = 1/2kx^2 comes from. If F=kx, then wouldnt it be kx*x = kx^2 ?
You should already know that the force that a spring exerts is proportional to the displacement (i.e. $$F=kx$$), and be familiar with the work-energy equation for a constant force ($$W=F d$$).

Now, we're not entirely sure about what the work that is done by the spring is, but we can approximate it. For example, we could cut the path of the spring into lots of little pieces, and approximate the work done on each of the peices of the path by picking the smallest force that the spring exerts on that segment to calculate the work for that segment. It turns out that it's possible to show that the total of these sengments tends toward $$\frac{1}{2}kx^2$$ as the segments get smaller and smaller.

russ_watters said:
To get you started, an integral is the area under a curve and the derivative is the slope of the curve. Its more complicated than that, of course...
Thanks for the tip.

NateTG said:
You should already know that the force that a spring exerts is proportional to the displacement (i.e. $$F=kx$$), and be familiar with the work-energy equation for a constant force ($$W=F d$$).