3.3 Forces and Potential Energy

Potential energy depends on force type

Potential energy (PE) is energy stored due to an object's position within a system. Gravitational PE near Earth's surface is *PEg = mgh*, valid because gravity is (approximately) constant. Elastic PE in an ideal spring is *PEs = ½kx²*, where x is displacement from equilibrium. Both arise from conservative forces — forces where work done doesn't depend on the path taken.

Force from a potential energy curve

For a conservative force, the force is related to the negative slope of the PE vs. position graph: F = −ΔPE/Δx. Where the PE curve is steep, the force is large; where the curve is flat (slope = 0), the force is zero — these flat points are equilibrium positions. This connects graphical analysis to force direction and magnitude.

Conservative vs. nonconservative forces

Conservative forces (gravity, springs, electric forces) allow energy to be fully recovered as PE — work done against them is path-independent and reversible. Nonconservative forces (friction, air resistance) dissipate energy as heat/sound, so work done depends on path, and no PE function can be defined for them.

Key terms

Gravitational potential energy

PEg = mgh; energy stored due to an object's height above a reference point.

Elastic potential energy

PEs = ½kx²; energy stored in a stretched or compressed spring.

Conservative force

A force where work done is path-independent (e.g., gravity, springs); allows a potential energy function to be defined.

Nonconservative force

A force where work done depends on path and mechanical energy is dissipated (e.g., friction, air resistance).

Force from PE graph

F = −ΔPE/Δx — the force equals the negative slope of the potential energy vs. position graph.

Try an AP-style question

A 2 kg box is lifted 5 m. What is its gain in gravitational PE? (g ≈ 10 m/s²)

1. 20 J
2. 100 J
3. 50 J
4. 10 J