# Gravitational potential energy question (that my teachers can't explain)

## Homework Statement

Hi everybody, this is a conceptual question about gravitational potential energy that my textbook and high school teacher seem to give incomplete answers.

The problem involves two masses $$m_{1}$$ and $$m_{2}$$, where $$m_{2}$$ > $$m_{1}$$, which are both at a radius R from the Earth's center. (The radius R is significantly large that Earth's gravitational field cannot be considered constant).

Question: Which mass has greater gravitational potential energy?

## Homework Equations

The gravitational potential energy is given by

$$E_{p} = -G \frac{mM_{E}}{r}$$

where $$M_{E}$$ is the mass of the Earth.

## The Attempt at a Solution

Since r=R for both masses, the only difference are the masses of the objects. Since $$m_{2}$$>$$m_{1}$$, the equation above would state that the gravitational potential energy (GPE) of $$m_{2}$$ will be more negative, hence less than the GPE for $$m_{1}$$.

i.e. $$GPE_{m_{1}}$$ > $$GPE_{m_{2}}$$

However this doesn't seem to make sense to me. Surely a bigger mass above the Earth will have more gravitational energy. The textbook answer tells me that the gravitational potential energy of $$m_{2}$$ is indeed larger, but it explains it using $$E_{p}$$ = mgh, where h is the height of the object.

Doc Al
Mentor
Surely a bigger mass above the Earth will have more gravitational energy.
Measured from what point?

There's no absolute significance to potential energy, only to changes in potential energy. It depends on your reference point. The equation you gave takes GPE = 0 at r = ∞. If you wanted to measure which mass has more GPE measured from the earth's surface, then you'd have to compute the change in GPE from r = Re to r = Re + h. Then you'll see that measured from that point, m2 has greater GPE than m1.

gneill
Mentor
The negative sign on the formula for potential energy is the result of the choice of placement of the origin for the PE measurement. The choice that was selected was to make PE zero when objects are separated by an infinite distance. As a consequence, the attractive force of gravity makes the work required to bring two gravitating bodies towards each other negative - you get energy out, which is why things accelerate when they fall.

As an analogy, suppose Ned and Fred are located at -20m and +7m from the origin along an x-axis. Does Ned have "less distance" from the origin than Fred because his coordinate is negative?

If you want to see that the larger mass has greater "potential" to do work, take the difference between the potentials at two different radii. If the difference is negative, you get energy out of the field, turning into kinetic energy for the mass. If the difference is positive, you had to do work to move the mass to the new position (you either lifted it, or its kinetic energy was "stolen", slowing it down).

So, don't be fooled by the sign attached to the number representing the potential energy; magnitude counts.