Potential Energy Formula
The gravitational potential energy formula PE = mgh calculates stored energy based on height.
Essential for energy conservation problems.
The Formula
Gravitational potential energy is the energy stored in an object due to its height above a reference point. The higher the object, the more potential energy it has.
Variables
| Symbol | Meaning |
|---|---|
| PE | Potential energy (measured in joules, J) |
| m | Mass of the object (measured in kilograms, kg) |
| g | Acceleration due to gravity (9.81 m/s² on Earth) |
| h | Height above the reference point (measured in meters, m) |
Example 1
A 5 kg book sits on a shelf 2.5 m above the floor. What is its potential energy?
Identify the values: m = 5 kg, g = 9.81 m/s², h = 2.5 m
Apply the formula: PE = mgh = 5 × 9.81 × 2.5
PE = 122.625 J (approximately 122.6 J)
Example 2
A roller coaster car has 294,300 J of potential energy at the top of a 30 m hill. What is the mass of the car?
Rearrange: m = PE / (gh)
m = 294,300 / (9.81 × 30) = 294,300 / 294.3
m = 1,000 kg
When to Use It
Use the potential energy formula when dealing with objects at height.
- Calculating stored energy before an object falls
- Energy conservation problems (PE converting to KE)
- Determining the height needed to achieve a certain energy
- Roller coasters, waterfalls, falling objects, and lifting problems
Key Notes
- Gravitational PE: PE = mgh: m is mass (kg), g ≈ 9.81 m/s² (surface gravity), h is height above the reference level (m). The reference level (h = 0) can be chosen at any convenient point — only changes in PE are physically meaningful.
- Elastic (spring) PE: PE = ½kx²: For a spring with constant k compressed or stretched by distance x. Unlike gravitational PE, elastic PE is always positive because it depends on x² — the spring stores energy whether stretched or compressed.
- Conservation of mechanical energy: In the absence of friction, KE + PE = constant. A falling object converts gravitational PE to kinetic energy. At the lowest point, PE is minimum and KE is maximum.
- PE is relative, not absolute: Only differences in PE matter — PE = mgh at a cliff edge means nothing until you define the reference height. Setting PE = 0 at the ground is conventional but not required.
- Gravitational PE at large distances: At heights comparable to Earth's radius, PE = −GMm/r (negative because it's attractive). Near the surface, the simpler PE = mgh is a linear approximation valid for h ≪ R_Earth.