3.3 Potential Energy

Potential energy is stored energy that belongs to a system whose objects interact through conservative forces like gravity or a spring. It is a scalar, you get to pick where zero is, and only changes in potential energy actually matter for the physics.

Why This Matters for the AP Physics 1 Exam

Potential energy is one of the core building blocks for energy conservation, which shows up across many units. Once you can describe how a system stores energy and write the right expression for it, you can track how energy moves between potential and kinetic forms in later problems.

This topic gives you the language and equations you need to reason through both multiple-choice questions and free-response problems. The first free-response question, the Mathematical Routines question, asks you to build and use mathematical models, justify claims with evidence, and explain a physical situation clearly. Spring and gravitational potential energy are good practice for that style of derivation and explanation, even though that question can draw from any unit.

Key Takeaways

• Potential energy belongs to a system of two or more objects that interact through conservative forces, not to a single isolated object.
• It is a scalar, so it has magnitude but no direction.
• You choose where potential energy equals zero to make a problem easier; only changes in potential energy have physical meaning.
• Elastic potential energy of an ideal spring: $U_s = \frac{1}{2}k(\Delta x)^2$, where $\Delta x$ is the stretch or compression from equilibrium.
• Gravitational potential energy between two spherical masses: $U_g = -G\frac{m_1 m_2}{r}$; the near-surface approximation is $\Delta U_g = mg\Delta y$.
• For a system with more than two objects, add the potential energy of every pair.

Potential Energy of Systems

A system of two or more objects has potential energy when the objects interact with each other through conservative forces. In AP Physics 1, potential energy is treated as belonging to the system, not to a single isolated object.

Conservative Forces

Conservative forces let a system store energy based on the relative positions of objects within it. When objects move under these forces, energy can be stored and later recovered.

Key characteristics of conservative forces:

• Work done by a conservative force is path-independent and depends only on the initial and final configurations.
• If the system returns to its starting configuration, the work done by the conservative force, and the change in potential energy, is zero.
• Potential energy is associated only with conservative forces.
• Examples in AP Physics 1 include gravitational forces and spring forces.

In contrast, nonconservative forces like friction and air resistance are path-dependent and dissipate mechanical energy as thermal energy or sound, so that energy is not stored in the configuration.

Scalar Nature of Potential Energy

Potential energy is a scalar quantity, so it has magnitude but no direction.

This has a few useful consequences:

• Potential energy depends on the relative positions of objects within the system.
• You do not break it into vector components.
• Only the configuration matters for the value, not the path taken to get there.

The scalar nature makes potential energy easier to use in energy conservation calculations.

Zero Potential Energy Definition

The zero point for potential energy is a choice the observer makes to simplify the analysis.

When working with potential energy:

• You can define zero potential energy at any convenient reference point.
• Ground level is a common zero for gravitational potential energy near Earth's surface.
• The spring's equilibrium position is the natural zero for elastic potential energy.
• Only changes in potential energy affect physical outcomes, not the absolute value.

Potential Energy Descriptions

Different systems store potential energy in specific ways, each with its own formula.

For an ideal spring, the elastic potential energy is:

• $U_s = \frac{1}{2}k(\Delta x)^2$
• $k$ is the spring constant (stiffness).
• $\Delta x$ is the distance stretched or compressed from equilibrium length.
• The quadratic form means energy grows with the square of the displacement.

For gravitational potential energy between two approximately spherical masses:

• $U_g = -G\frac{m_1 m_2}{r}$
• $G$ is the universal gravitational constant.
• $m_1$ and $m_2$ are the masses.
• $r$ is the distance between their centers.
• The negative sign reflects that gravity is attractive.

Near a planet's surface, where the gravitational field is nearly constant, the change in gravitational potential energy simplifies to:

• $\Delta U_g = mg\Delta y$
• $g$ is the local gravitational field strength (about 9.8 m/s² near Earth).
• $\Delta y$ is the height change relative to a reference point.
• This linear form applies when the height change is small compared to the planet's radius.

Total Potential Energy Calculation

When a system contains more than two objects, the total potential energy is the sum of the potential energy of each pair.

The process:

1. Identify all pairs of objects in the system.
2. Calculate the potential energy for each pair using the appropriate equation.
3. Add the contributions from all pairs.

For a three-object system (A, B, and C), the total potential energy is: $U_{total} = U_{AB} + U_{AC} + U_{BC}$

This pairwise approach lets you analyze complex systems, such as several masses interacting gravitationally, by breaking them into simpler interactions.

How to Use This on the AP Physics 1 Exam

Problem Solving

• Match the system to the right formula. A stretched or compressed spring uses $U_s = \frac{1}{2}k(\Delta x)^2$. Two distant spherical masses use $U_g = -G\frac{m_1 m_2}{r}$. A small height change near a surface uses $\Delta U_g = mg\Delta y$.
• State your zero reference before plugging in. This keeps your sign conventions consistent for the rest of the problem.
• Watch units. Spring constant is in N/m, masses in kg, distances in m, and energy in joules.

Free Response

• When you derive an expression, define your system and your zero of potential energy first, then justify each step with a physical principle.
• For multi-object systems, show that you summed every pair rather than treating each object separately.
• Keep your explanation organized and sequential, citing the conservative force involved and what configuration sets your reference.

Common Trap

• Do not attach potential energy to a single object. It belongs to the interacting system.

Practice Problem 1: Spring Potential Energy

Solution

Use the elastic potential energy formula: $U_s = \frac{1}{2}k(\Delta x)^2$

Substitute the given values: $U_s = \frac{1}{2} \times 150 \text{ N/m} \times (0.25 \text{ m})^2$ $U_s = \frac{1}{2} \times 150 \times 0.0625$ $U_s = 4.69 \text{ J}$

The spring stores about 4.69 joules of potential energy when stretched 0.25 m from equilibrium.

Practice Problem 2: Gravitational Potential Energy

Solution

Because the height change is small compared to Earth's radius, use the near-surface approximation: $\Delta U_g = mg\Delta y$

Substitute the given values: $\Delta U_g = 60 \text{ kg} \times 9.8 \text{ m/s}^2 \times (2500 - 1200) \text{ m}$

$\Delta U_g = 60 \times 9.8 \times 1300$ $\Delta U_g = 764{,}400 \text{ J} \text{ or } 764.4 \text{ kJ}$

The hiker's gravitational potential energy increases by about 764.4 kilojoules during the climb.

Practice Problem 3: Total Potential Energy

Solution

The total potential energy here comes only from the spring, since there is no height change on a horizontal surface.

Use the spring potential energy formula: $U_s = \frac{1}{2}k(\Delta x)^2$ $U_s = \frac{1}{2} \times 100 \text{ N/m} \times (0.3 \text{ m})^2$ $U_s = \frac{1}{2} \times 100 \times 0.09$ $U_s = 4.5 \text{ J}$

The total potential energy of the system is 4.5 joules, stored entirely in the stretched spring.

Common Misconceptions

• A single isolated object does not have potential energy. Potential energy belongs to a system of two or more objects interacting through conservative forces.
• Choosing a different zero point does not change the physics. Only differences in potential energy matter, so absolute values are not physically meaningful.
• The negative sign in $U_g = -G\frac{m_1 m_2}{r}$ does not mean negative energy is stored in a strange way; it reflects that gravity is attractive and that potential energy increases toward zero as the masses move far apart.
• Friction and air resistance are nonconservative, so the energy they remove is not stored as potential energy. It is dissipated as thermal energy or sound.
• $\Delta U_g = mg\Delta y$ is only an approximation for small height changes near a surface. For large distances between masses, use $U_g = -G\frac{m_1 m_2}{r}$.
• In $U_s = \frac{1}{2}k(\Delta x)^2$, $\Delta x$ is measured from the spring's equilibrium length, not from any random starting point.

Related AP Physics 1 Guides

• 3.2 Work
• 3.4 Conservation of Energy
• 3.5 Power
• 3.1 Translational Kinetic Energy