In Gravitational Potential Energy you met $PE = mgh$. Now look closely at where that energy lives. It isn't stored in the lifted book alone; it belongs to the book and Earth together, because it depends on how far apart they are.
Key idea
For an object–Earth system near Earth's surface, $\Delta U_g = mg\,\Delta h$. Only changes in $U_g$ matter, so you may put $U_g = 0$ at any convenient height. $\Delta U_g$ depends only on the change in height, not on the path.
Why it works
Lift a book at steady speed and you do work $mg\,\Delta h$ against gravity, but the book's kinetic energy doesn't change. The energy went into the separation between the book and Earth. Gravity's work on any path depends only on how far up or down you end, because only the vertical parts of the path count. So the stored energy is path-independent, and that is what makes it a potential energy.
Worked examples
Worked example: Up the stairs
A $55\ \text{kg}$ student climbs to a floor $3.5\ \text{m}$ higher. Find $\Delta U_g$ for the student–Earth system.
- $\Delta U_g = mg\,\Delta h = (55)(9.8)(3.5)$.
- $\Delta U_g \approx 1{,}890\ \text{J}$.
Answer: about $1{,}890\ \text{J}$
Worked example: Path does not matter
A $20\ \text{kg}$ cart is pushed up a winding ramp $30\ \text{m}$ long to a platform $4\ \text{m}$ higher. Find $\Delta U_g$.
- Only the change in height counts: $\Delta U_g = (20)(9.8)(4) = 784\ \text{J}$.
- The $30\ \text{m}$ ramp length is a distractor.
Answer: $784\ \text{J}$
Worked example: Choosing the zero
A $2\ \text{kg}$ box on a shelf $1.8\ \text{m}$ above the floor, next to a $0.8\ \text{m}$ table. Find $U_g$ with $U_g = 0$ at the floor, then at the tabletop.
- Floor as zero: $U_g = (2)(9.8)(1.8) = 35.3\ \text{J}$.
- Tabletop as zero: $U_g = (2)(9.8)(1.0) = 19.6\ \text{J}$.
- Different values, but moving the box from the shelf to the floor changes $U_g$ by $-35.3\ \text{J}$ in both choices.
Answer: $35.3\ \text{J}$ or $19.6\ \text{J}$, depending on the zero
Common mistake
Thinking the potential energy depends on the path taken between two heights: that a long gentle ramp stores more energy than a short steep one. The ramp takes a smaller force over a longer distance, and the work against gravity comes out the same. Check: $\Delta U_g$ has only $\Delta h$ in it, never a path length.