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

• Mugged
In summary, the potential energy of a pendulum is not invariant with respect to coordinates, but it is still conserved.
Mugged

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 what's the distinction? what is the actual potential energy? why have i seen it done both ways?

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?

suppose i wanted to know the energy of the system?

Mugged said:
suppose i wanted to know the energy of the system?

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.

Mugged said:
suppose i wanted to know the energy of the system?
You can know it in respect to the origin you choose.

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

shouldnt the energy be invariant with respect to coordinates?

Mugged said:
shouldnt the energy be invariant with respect to coordinates?
No, energy is definitely not invariant. It is conserved, not invariant. Those are two different concepts.

Last edited:
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!

Mugged said:
shouldnt the energy be invariant with respect to coordinates?

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:
maajdl said:
It 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.
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.

## What is potential energy?

Potential energy is the energy that an object possesses due to its position or state. It is the energy that an object has the potential to convert into other forms of energy.

## What is the potential energy of a pendulum?

The potential energy of a pendulum is the energy that it possesses due to its position above the equilibrium point. It is directly proportional to the height at which the pendulum is raised and the mass of the pendulum.

## How is the potential energy of a pendulum calculated?

The potential energy of a pendulum can be calculated using the formula PE = mgh, where m is the mass of the pendulum, g is the acceleration due to gravity, and h is the height of the pendulum above the equilibrium point.

## What is the significance of placing the datum in potential energy calculations?

The datum is the reference point from which the height of the pendulum is measured. Placing the datum at the equilibrium point of the pendulum ensures that the potential energy calculated is only due to the height of the pendulum and not its position.

## Can the potential energy of a pendulum be negative?

Yes, the potential energy of a pendulum can be negative if the pendulum is below the equilibrium point. This indicates that the pendulum has lost energy and has converted it into kinetic energy.

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