# 2 Different Ways to Write Potential Energy of a Pendulum

• zachx

#### zachx

I have been doing pendulum problems lately, and I have found 2 different formulations for potential energy of a pendulum.

U=mgl(1-cos(Θ)) and U=-mglcos(Θ)

The first says U=0 when Θ=0 (at the bottom). The second has U=0 when Θ=π/2 (halfway to the top).

Both give the same equation of motion, but the Lagrangians are different.

Which is better/conventional?

Either way is fine. I would tend to use the second just because it is easier to write. Nobody would object to you using the other one if you prefer.

zachx and vanhees71
Just another remark. The Lagrangian is not a physical observable but it's used to define the action for Hamilton's action principle. Thus two Lagrangians ##L(q,\dot{q},t)## and ##L'(q,\dot{q},t)## are completely equivalent if
$$L'(q,\dot{q},t)=L(q,\dot{q},t)+\frac{\mathrm{d}}{\mathrm{d} t} \Omega(q,t)=L(q,\dot{q},t)+\dot{q} \cdot \vec{\nabla}_q \Omega(q,t) + \partial_t \Omega(q,t),$$
because adding such a term leaves the variation of the action and thus the Euler-Lagrange equations invariant. This is a very important concept for the derivation of Noether's theorem.

zachx and Dale

## 1. What is potential energy?

Potential energy is the energy that an object possesses due to its position or configuration. In the case of a pendulum, potential energy is the energy that is stored in the system when the pendulum is lifted to a certain height.

## 2. How is potential energy calculated for a pendulum?

There are two ways to calculate potential energy for a pendulum. The first method is 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. The second method is using the formula PE = (1/2)kx^2, where k is the spring constant of the pendulum and x is the displacement from equilibrium.

## 3. What is the difference between the two ways of writing potential energy for a pendulum?

The first method, PE = mgh, is based on the gravitational potential energy of the pendulum, while the second method, PE = (1/2)kx^2, is based on the elastic potential energy of the pendulum. The first method is more commonly used for simple pendulums with a mass attached to a string, while the second method is used for more complex pendulums with a spring system.

## 4. Which method should be used for calculating potential energy of a pendulum?

The method used to calculate potential energy of a pendulum depends on the specific situation and the type of pendulum being analyzed. Both methods are valid and can be applied, but the appropriate method should be chosen based on the physical properties of the pendulum.

## 5. How does potential energy affect the motion of a pendulum?

Potential energy plays a crucial role in the motion of a pendulum. As the pendulum swings, potential energy is constantly being converted into kinetic energy and vice versa. This exchange of energy is what causes the pendulum to continue swinging back and forth. The maximum potential energy of the pendulum occurs at the highest point of its swing, while the minimum potential energy occurs at the lowest point of its swing.