Potential Energy | AP Physics 1 Unit 3 Study Guide

Getting Started

Imagine lifting a heavy book from the floor to a high shelf or drawing back the string of a bow. In both cases, you do work, and the energy you expend seems to be stored, ready to be released. This "stored energy" is not located in a single object but within the system of interacting objects—the book and the Earth, or the bow and its string. This chapter explores potential energy, a way to quantify the energy stored in a system due to the specific arrangement or configuration of its parts.

What You Should Be able to Do

After completing this section, you will be able to:

Describe potential energy as a property of a system of interacting objects, not a single object.
Explain that potential energy only exists for systems where objects interact through conservative forces, like gravity or ideal springs.
Calculate the change in gravitational potential energy for an object near a planet's surface.
Calculate the elastic potential energy stored in an ideal spring.
Justify the choice of a "zero point" for potential energy in a given problem.

Key Concepts & Mechanisms

System & Preconditions

Potential energy is fundamentally about interactions within a system. A system is the collection of objects we choose to analyze. For potential energy to be a useful concept, the objects within the system must interact through conservative forces. A conservative force is an interaction where the work done to move an object between two points is independent of the path taken. The two primary examples you will encounter are the force of gravity and the force exerted by an ideal spring.

The key precondition for a system to have potential energy is the presence of an internal conservative force that depends on the position or configuration of the objects within that system. For example, the gravitational force between a satellite and the Earth depends on their separation distance. The force from a spring depends on how much it is stretched. Because of this, the energy "stored" in the system changes as the objects' positions change. Potential energy is a scalar quantity, meaning it has magnitude but no direction, and its standard unit is the Joule (J).

Key Steps / Relations

Define the System: First, clearly identify the boundaries of your system. Is it a ball and the Earth? Is it a block and a spring? Potential energy is a shared property of the interacting objects.
Identify the Conservative Interaction: Confirm that the interaction is conservative. Near the Earth's surface, gravity is treated as a conservative force. The force from an ideal spring is also conservative. Forces like friction and air resistance are non-conservative, and we cannot define a potential energy associated with them.
Establish a Reference Point (Zero Level): A crucial feature of potential energy is that its absolute value is meaningless; only changes in potential energy matter. We must define a position or configuration where the potential energy is zero. This choice is arbitrary and should be made to simplify the problem. For gravity, this is often the ground or the lowest point in the object's path. For a spring, the zero point is always chosen to be its natural, equilibrium length.
Relate Configuration to Potential Energy: Use the established formulas to connect the system's configuration to its potential energy relative to the zero point.
- Gravitational Potential Energy: For a system consisting of an object of mass $m$ and a planet with a uniform gravitational field $g$ (a valid approximation near the surface), the change in potential energy is related to the change in vertical position, $Δ y$ .
  $Δ U_{g} = m g Δ y$
  Here, $Δ U_{g}$ is the change in gravitational potential energy (in Joules), $m$ is the mass (in kg), $g$ is the gravitational field strength (approx. 9.8 N/kg on Earth), and $Δ y$ is the change in vertical height (in meters).
- Elastic Potential Energy: For a system including an ideal spring with spring constant $k$ , the potential energy stored when it is stretched or compressed by a displacement $Δ x$ from its equilibrium position is:
  $U_{s} = \frac{1}{2} k (Δ x)^{2}$
  Here, $U_{s}$ is the elastic potential energy (in Joules), $k$ is the spring constant (in N/m), and $Δ x$ is the displacement from equilibrium (in meters). Note that because $Δ x$ is squared, the potential energy is always positive, whether the spring is stretched or compressed.

Outputs & Effects

A change in the configuration of a system (e.g., lifting an object, compressing a spring) causes a change in its potential energy. This stored energy can be converted into other forms. For instance, if you release a ball held at a height, the gravitational potential energy of the Earth-ball system is converted into the ball's kinetic energy (the energy of motion) as it falls. If only conservative forces do work within an isolated system, the total mechanical energy (the sum of kinetic and potential energies) remains constant.

Regulation & Limits

The models for potential energy rely on important assumptions.

The equation $Δ U_{g} = m g Δ y$ is an approximation that is only valid when the object remains close to the surface of the planet, where the gravitational field $g$ can be considered constant. It does not apply to satellites in high orbit.
The equation $U_{s} = \frac{1}{2} k (Δ x)^{2}$ applies only to "ideal" springs that obey Hooke's Law and return to their original shape after being deformed. It does not account for energy lost as thermal energy if a spring is stretched beyond its elastic limit.

Key Models & Diagrams

The two primary models for potential energy in this course can be linked directly from the type of interaction to the relevant equation.

Interaction Type	System Configuration	Potential Energy Equation
Gravitational (near-surface)	An object's vertical position ( $y$ ) relative to a chosen zero level in a uniform gravitational field.	$Δ U_{g} = m g Δ y$
Elastic (ideal spring)	A spring's displacement ( $Δ x$ ) from its natural, equilibrium length.	$U_{s} = \frac{1}{2} k (Δ x)^{2}$

Key Components & Evidence

System: A collection of interacting objects chosen for analysis (e.g., Earth-ball system).
Potential Energy ( $U$ ): The energy stored within a system due to the relative positions of its constituent parts. Its SI unit is the Joule (J).
Conservative Force: An interaction force (like gravity) where the work done moving between two points is path-independent, allowing for the definition of potential energy.
Gravitational Potential Energy ( $U_{g}$ ): The energy stored in the gravitational interaction between two objects, such as a planet and a smaller mass.
Elastic Potential Energy ( $U_{s}$ ): The energy stored in a deformable object, like a spring, when it is stretched or compressed.
Mass ( $m$ ): The property of an object that measures its inertia. Its SI unit is the kilogram (kg).
Gravitational Field Strength ( $g$ ): The gravitational force exerted per unit mass on an object. Near Earth's surface, its value is approximately 9.8 N/kg.
Spring Constant ( $k$ ): A measure of a spring's stiffness, representing the force required to stretch it per unit length. Its SI unit is the Newton per meter (N/m).
Vertical Position ( $y$ ): The height of an object relative to a defined origin in a coordinate system. Its SI unit is the meter (m).
Displacement from Equilibrium ( $Δ x$ ): The distance a spring is stretched or compressed from its relaxed, natural length. Its SI unit is the meter (m).

Skill Snapshots

Causation

The work done against the conservative gravitational force when lifting a book causes an increase in the gravitational potential energy of the book-Earth system.
Pushing a block to compress a spring causes an increase in the elastic potential energy stored in the spring.
The presence of an internal conservative force that depends on position causes a system to possess potential energy that can be converted to kinetic energy.

Comparison

Gravitational potential energy is associated with the separation between massive objects, while elastic potential energy is associated with the deformation of an object.
The zero point for gravitational potential energy is an arbitrary choice made by the observer (e.g., the floor), whereas the zero point for an ideal spring's potential energy is non-arbitrarily defined at its equilibrium position.
A system's gravitational potential energy can be positive, negative, or zero depending on the reference point, while the elastic potential energy of an ideal spring is always non-negative ( $U_{s} \geq 0$ ).

Change Over Time

Baseline State: A pendulum bob hangs motionless at the lowest point of its swing. We define the gravitational potential energy of the bob-Earth system to be zero at this height.
Change 1: As the bob swings upward, its height increases, and its speed decreases. The system's gravitational potential energy increases as its kinetic energy decreases.
Change 2: As the bob swings back down through the lowest point, its height decreases, and its speed increases. The system's gravitational potential energy is converted back into kinetic energy.
Continuity: If we ignore air resistance and friction, the total mechanical energy of the system (the sum of its kinetic and potential energies) remains constant throughout the swing.

Common Misconceptions & Clarifications

Misconception: A single object, like a ball in the air, "has" gravitational potential energy.
Clarification: Potential energy is a property of a system of interacting objects. The ball has gravitational potential energy because of its interaction with the Earth. It is more accurate to speak of the potential energy of the "ball-Earth system."
Misconception: Negative potential energy is impossible or means the system is missing energy.
Clarification: Potential energy is a relative quantity. A negative value simply means the system has less potential energy than it does at the location defined as the zero reference point. For example, if you set the zero level for $U_{g}$ at your tabletop, a book on the floor has negative potential energy relative to the table.
Misconception: The formula for gravitational potential energy is always $U_{g} = m g y$ .
Clarification: The physically significant quantity is the change in potential energy, $Δ U_{g} = m g Δ y$ . The simpler equation $U_{g} = m g y$ is a direct consequence of this, but it is only true if you have explicitly defined the potential energy to be zero ( $U_{g} = 0$ ) at the height $y = 0$ .
Misconception: The variable $Δ x$ in the elastic potential energy formula is the total length of the spring.
Clarification: The variable $Δ x$ represents the displacement—how far the spring has been stretched or compressed away from its natural, relaxed length (its equilibrium position). A spring that is neither stretched nor compressed has $Δ x = 0$ and stores zero elastic potential energy.

One-Paragraph Summary

Potential energy is the energy stored within a system of objects due to their physical arrangement or configuration. This form of energy is a scalar quantity measured in Joules and is only associated with internal conservative forces, such as gravity and the elastic force of an ideal spring. The absolute value of potential energy is relative, depending on an arbitrarily chosen zero reference point, making the change in potential energy the most important physical quantity. The change in gravitational potential energy near a planet's surface is given by $Δ U_{g} = m g Δ y$ , while the elastic potential energy in a spring is $U_{s} = \frac{1}{2} k (Δ x)^{2}$ . Understanding potential energy is essential for applying the principle of conservation of energy, which states that energy can be transformed from one form to another (e.g., potential to kinetic) but is not lost.

Potential Energy - AP Physics 1: Algebra-Based Study Guide