William asked in Science & MathematicsPhysics · 7 years ago

# EP = -GMm/r?

Why is it negative when you aren't looking for a negative value?

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• 7 years ago

When we define PE --> 0 as r --> r, such as when calculating escape velocity from a planet with mass M, we must still observe the physics of stored energy in that we store more energy as we move something farther away from the gravity source M. That is, the work we do is converted into stored energy.

But look what happens in PE = GMm/r, the stored energy gets smaller as r --> infinity. And that's just not the physics of energy is it? So what to do?

Redefine PE = 0 as being the most positive value of the stored energy and that value is approached for very very large r. Which puts any PE at less than the uncountably large r in the negative domain. So with the redefined most positive PE, we must have PE = - mGM/r for any r less than infinity, which is to say for any r as there is no bound on infinity.

We run into this redefined PE < 0 when solving for escape velocity from a planet of mass M and radius r = R. In which case KE + PE = 0 is the assume total energy once the escape has been made. So 1/2 mU^2 - mGM/R = 0 and U = sqrt(2GM/R) is the escape velocity from a planet with mass M and radius R.

• RickB
Lv 7
7 years ago

Potential energy is energy of position, which means it's always measured REALTIVE to some arbitrary "zero" position. For example, you may decided that the "zero" position corresponds to ground level; in that case a rock on your roof has positive PE; but a rock in your basement has negative PE.

Positive PE really just means, the rock can do work on other objects as it moves from the roof to the zero position (the move liberates energy). Negative PE just means, you have to do work ON the rock to lift it from the basement to the zero level (the move CONSUMES energy).

When looking at gravitational PE for locations that are not necessarily close to the surface of earth (i.e. where the force of gravity may vary a lot, or may involve several gravitating bodies), it's convenient for mathematical reasons to choose "the zero position" to be infinitely far away. That means no matter where you are, you're in "the basement," and you need to consume energy to lift a rock up to the zero position (which is infinitely far over your head). Because of this, the sign of the PE is always negative.