Gravitational potential energy U
● VERBATIM — examined word-for-wordGravitational potential energy at a point is the work done by an external force in bringing a mass from infinity to that point, without change in kinetic energy. Scalar, unit J.
U = mφ = −GMm/rWhat this actually means
Same sentence as potential, minus 'per unit mass' and with a specific mass instead of a test mass. If you can say one you can say the other, so learn them as a pair and keep the difference sharp.
U is negative for the same reason φ is: U = 0 at infinity by definition, and gravity does positive work as the mass falls inwards, so the store of energy drops below zero. Dropping the minus sign here is the single most expensive habit in the topic.
U belongs to the SYSTEM of two masses, not to the small one alone. In H2 you can speak loosely about 'the satellite's GPE', but if a question asks whose energy it is, it is the Earth-satellite system's.
Near the Earth's surface U ≈ mgh, but that is a special case with the zero shifted to ground level and g assumed constant. The two zeros are different, so never mix a mgh answer with a −GMm/r answer in the same calculation.
In energy questions always work with CHANGES: ΔU = U_final − U_initial = (−GMm/r_f) − (−GMm/r_i). Subtract in that order and the sign of the answer tells you which way the mass moved.
Quoting U as positive, or reusing mgh over heights comparable with the Earth's radius.
Prove it — watch it be true
- Set the work-done two-marker tool with marker A far out and marker B close to the planet.
- Read the work done and confirm it is negative, meaning the external force did negative work bringing the mass in.
- Check the U readout equals m × φ at the same point by changing the mass and watching U scale while φ stays fixed.