1. Definition

Potential Energy is energy stored in a system due to the position or configuration of objects.

Unlike kinetic energy, which depends on motion, potential energy depends on where objects are located relative to one another or how they are arranged.

Potential energy exists only for conservative forces.

2. Conservative Forces

A force is conservative if the work it does depends only on the initial and final positions, not on the path taken.

Examples:

• gravitational force
• spring force

For conservative forces, potential energy can be defined.

3. Relationship Between Work and Potential Energy

For a conservative force:

\(W_c=-\Delta U\)

where:

• \(W_c\)​ = work done by the conservative force
• \(U\) = potential energy

The negative sign means:

• when potential energy decreases, the force does positive work
• when potential energy increases, the force does negative work

4. Gravitational Potential Energy

Near Earth’s surface:

\(U_g=mgh\)

where:

• m = mass
• $g$ = gravitational acceleration
• $h$ = height relative to a chosen reference level

This expression is valid when $g$ is approximately constant.

Physical Meaning

Higher elevation corresponds to greater gravitational potential energy.

If an object falls:

• \(h\) decreases
• \(U_g\)​ decreases
• kinetic energy increases

5. Reference Level

Potential energy is always measured relative to a chosen zero level.

Only changes in potential energy matter:

\(\Delta U=U_f-U_i\)​

The choice of zero does not affect physical predictions.

6. Elastic Potential Energy

A stretched or compressed spring stores elastic potential energy.

The spring force follows:

\(F=-kx\)

where:

• k = spring constant
• x = displacement from equilibrium

Potential energy stored in the spring:

\(U_s=\frac{1}{2}kx^2\)

7. Force and Potential Energy

For conservative forces:

\(F=-\frac{dU}{dx}\)​

This equation shows that force is related to the slope of the potential-energy function.

A force points toward decreasing potential energy.

8. Energy Diagrams

Potential-energy graphs provide information about motion.

Important ideas:

Minimum Potential Energy

Objects tend to move toward stable equilibrium positions where potential energy is lowest.

Maximum Potential Energy

Often corresponds to unstable equilibrium.

Slope of the Graph

\(F=-\frac{dU}{dx}\)

A steeper slope indicates a larger force.

9. Conservation of Mechanical Energy

When only conservative forces act:

\(K+U=constant\)

or

\(K_i+U_i=K_f+U_f\)​

Mechanical energy is conserved.

10. Examples

Falling Object

As an object falls:

\(U_g \downarrow\)

\(K\ uparrow\)

Potential energy is converted into kinetic energy.

Tossed Ball

As a ball rises:

\(K \downarrow\)

\(U_g \uparrow\)

At the highest point:

• kinetic energy is minimum
• potential energy is maximum

Spring Compression

Compressing a spring stores energy:

\(U_s=\frac{1}{2}kx^2\)

When released, this energy becomes kinetic energy.

Summary

Potential energy is stored energy associated with position or configuration.

Gravitational potential energy:

\(U_g=mgh\)

Elastic potential energy:

\(U_s=\frac{1}{2}kx^2\)

Relationship to work:

\(W_c=-\Delta U\)

Key ideas:

• defined only for conservative forces
• depends on position, not motion
• can be converted into kinetic energy
• essential for applying conservation of mechanical energy

Potential energy provides a powerful alternative to force-based analysis and is one of the central concepts in AP Physics C Mechanics.