# Potential energy of a pendulum and where you place the datum.

1. ### Mugged

104

When you describe the potential energy PE = mgh, you must decide where to measure your h from. Throughout my years I've seen it measured from the mass to the 0 equilbrium point where you'd get that PE = mgL*(1-cosΘ) and also measured from the mass to the horizontal position where Θ=π/2 where you would get PE = -mgLcosΘ. the signs are with respect to the positive y-axis pointing up.

These are clearly not the same number, so whats the distinction? what is the actual potential energy? why have i seen it done both ways?

2. ### maajdl

379
It does not matter where you chose the reference point for potential energy.
Try to solve th problem with and arbitrary point of reference,
and observe that you always end up with the same solution.

Can you see why?

3. ### Mugged

104
suppose i wanted to know the energy of the system?

4. ### maajdl

379
You would never be interested to know that.
You would only like to know how much energy could be released
if the pendulum falls from one place to another.

5. ### nasu

You can know it in respect to the origin you choose.

6. ### Mugged

104
shouldnt the energy be invariant with respect to coordinates?

### Staff: Mentor

No, energy is definitely not invariant. It is conserved, not invariant. Those are two different concepts.

Last edited: Mar 18, 2014
8. ### UltrafastPED

1,918
When you calculate forces from a potential it goes: F = - gradient(potential energy).

You will note that shifting the potential energy by any constant amount does not change the force ... hence the dynamics is not affected by the choice of origin for a potential.

Energy is still conserved ... just don't change your origin partway through a calculation!

9. ### maajdl

379
It (the potential energy) is invariant with respect to the coordinate system.
But it depend on the reference point chosen.
When you change the system of coordinate, the coordinates of the reference point are also changed.
The coordinates used do not matter.

Last edited: Mar 18, 2014

### Staff: Mentor

The coordinates do matter, energy is not invariant with respect to the coordinate system.

I understand your point. You are distinguishing between coordinate system and reference point. It is a tenuous distinction since you can always consider h to be a coordinate, however, even accepting the distinction the fact remains that energy does depend on the coordinate system.

Consider kinetic energy. If you are sitting in a car then in a coordinate system attached to the car your KE is 0, but in a coordinate system attached to the ground your KE is non-0. Energy therefore does depend on the coordinates.