AP Physics 1 · Unit 3 · Topic 3.3
Potential Energy
Potential energy belongs to a system of interacting objects, not to an isolated object by itself. It records how the system's energy depends on configuration: separation in a gravitational interaction or deformation in an elastic interaction.
Learning Goals
After this lesson, you should be able to:
- identify the objects and conservative interaction that give a system potential energy;
- explain why the zero of potential energy is a convenient reference rather than a physical location;
- connect conservative-force work with potential-energy change;
- calculate near-surface and universal gravitational potential energy;
- calculate ideal-spring elastic potential energy;
- compare configurations using equations and graphs; and
- add pairwise potential-energy contributions in a many-object system.
1. Potential Energy Belongs to a System
A single modeled object can have kinetic energy, but potential energy requires an interaction among objects inside the chosen system. An object–Earth system can have gravitational potential energy, and an object–spring system can have elastic potential energy.
Saying “the object has \(mgh\)” is convenient shorthand. The more precise statement is that the object–planet system has gravitational potential energy because its members interact and have a particular separation.
2. Conservative Work and Energy Change
For a conservative interaction, the work done by the conservative force is the negative of the system's potential-energy change:
If gravity does positive work while an object falls, the gravitational potential energy decreases. If an external agent slowly lifts the object, the agent transfers energy into the system and its gravitational potential energy increases.
3. Configuration and Potential-Energy Graphs
4. Choosing the Zero Level
The observer may assign \(U=0\) at any convenient configuration. Adding the same constant to every potential-energy value changes no prediction:
A value of \(U=-20\,\mathrm J\) is allowed; it means the configuration lies \(20\,\mathrm J\) below the selected zero. Only differences in potential energy enter an energy change.
5. Gravity Near a Planet's Surface
Over heights small compared with a planet's radius, the gravitational field magnitude \(g\) is approximately constant:
If the zero is selected at \(y=0\), this becomes \(U_g=mgy\). The result depends only on the vertical-coordinate change, not on whether the object follows stairs, a ramp, or another path between the same endpoints.
6. Worked Example: Change in Height
Question: A \(2.5\,\mathrm{kg}\) backpack moves from \(y_i=1.2\,\mathrm m\) to \(y_f=4.8\,\mathrm m\). Use \(g=9.8\,\mathrm{m/s^2}\). Find the gravitational potential-energy change.
Answer: The backpack–Earth system gains about \(88\,\mathrm J\) of gravitational potential energy. Choosing a different zero changes both endpoint values but not this difference.
7. Universal Gravitational Potential Energy
For two approximately spherical masses separated center to center by distance \(r\), choose \(U_G=0\) at infinite separation:
The negative sign represents an attractive bound interaction under this zero convention. Increasing separation makes \(U_G\) less negative, so external energy must be supplied.
8. Worked Example: Increase Orbital Separation
Question: A small mass \(m\) moves slowly from distance \(R\) to \(2R\) from a spherical planet of mass \(M\). Find the change in gravitational potential energy.
Answer: The potential energy increases by \(GMm/(2R)\). It rises from a more negative value to a less negative value; it does not become positive.
9. Ideal-Spring Potential Energy
For an ideal spring obeying Hooke's law, let \(x\) be deformation measured from the relaxed length:
Because \(x\) is squared, equal compressions and extensions store equal energy. Doubling the deformation multiplies the elastic potential energy by four.
10. Worked Example: Releasing a Compressed Spring
Question: A spring with \(k=250\,\mathrm{N/m}\) changes from \(x_i=-0.12\,\mathrm m\) to \(x_f=-0.06\,\mathrm m\). Find \(\Delta U_s\) and the work done by the spring.
Answer: The elastic energy decreases by \(1.35\,\mathrm J\), while the spring does \(+1.35\,\mathrm J\) of work. Do not remove the square because the deformation is negative.
11. Reading and Comparing Graphs
| Model | Graph and dependence | Useful comparison |
|---|---|---|
| Near-surface gravity | \(U_g\) vs. \(y\): straight line | Twice \(\Delta y\) gives twice \(\Delta U_g\) |
| Ideal spring | \(U_s\) vs. \(x\): upward parabola | Twice \(\lvert x\rvert\) gives four times \(U_s\) |
| Universal gravity | \(U_G\) vs. \(r\): negative curve approaching zero | Twice \(r\) gives half the magnitude \(\lvert U_G\rvert\) |
A steeper change in \(U\) over the same position interval corresponds qualitatively to a stronger conservative interaction. AP Physics 1 questions can be answered with slopes, ratios, and algebraic comparisons without calculus.
12. Several Interacting Objects
For more than two masses, add the gravitational potential energy of every distinct pair exactly once:
For three masses, the pairs are \((1,2)\), \((1,3)\), and \((2,3)\). There is no separate three-object term in this model, and counting both \((1,2)\) and \((2,1)\) would double-count the same interaction.
13. Model Selection
| Situation | Appropriate model | Required reference |
|---|---|---|
| Small height change near one planet | \(\Delta U_g=mg\Delta y\) | Any consistent height zero |
| Large change in distance from a spherical body | \(U_G=-Gm_1m_2/r\) | Usually \(U=0\) as \(r\to\infty\) |
| Ideal spring deformation | \(U_s=\tfrac12kx^2\) | Usually \(U_s=0\) at \(x=0\) |
Do not use \(mgh\) for changes comparable to a planet's radius, and measure spring deformation from the relaxed length rather than from an arbitrary coordinate origin.
14. Investigation: Map a Spring's Energy
Use an ideal spring, force sensor, meterstick, and several hanging masses or a slow pulling mechanism.
- Locate the relaxed length and define it as \(x=0\).
- Measure force magnitude for several positive and negative deformations within the elastic range.
- Plot force magnitude against \(\lvert x\rvert\) and use the best-fit slope to estimate \(k\).
- Calculate \(U_s=\tfrac12kx^2\) for each configuration.
- Plot \(U_s\) against \(x\) and against \(x^2\).
- Explain deviations using force-sensor zero, length uncertainty, spring mass, hysteresis, or exceeding the ideal range.
The \(U_s\)-versus-\(x\) plot should be symmetric and curved, while \(U_s\)-versus-\(x^2\) should be approximately linear.
15. Common Traps and Key Takeaways
- Potential energy belongs to an interacting system, not to one isolated object.
- The zero is chosen; negative \(U\) is physically valid and only \(\Delta U\) affects predictions.
- Use \(W_c=-\Delta U\), so positive conservative work means decreasing potential energy.
- In \(mg\Delta y\), use vertical change, not path length.
- In \(\tfrac12kx^2\), measure \(x\) from relaxed length and square its sign.
- Use center-to-center separation in \(-Gm_1m_2/r\).
- For many objects, add each distinct pair once.
A \(0.50\,\mathrm{kg}\) block is raised from \(y=1.0\,\mathrm m\) to \(y=3.5\,\mathrm m\), then used to compress an ideal \(80\,\mathrm{N/m}\) spring by \(0.20\,\mathrm m\). Calculate the changes in gravitational and elastic potential energy. Find their sum and explain whether choosing \(U_g=0\) at \(y=1.0\,\mathrm m\) instead of at the floor changes any of these energy changes.
Concept scope cross-checked with the College Board AP Physics 1 Course and Exam Description and the open educational references OpenStax University Physics: Potential Energy of a System, OpenStax College Physics for AP Courses: Gravitational Potential Energy, and OpenStax College Physics for AP Courses: Conservative Forces and Potential Energy. Explanations, diagram, examples, checkpoint, and investigation are independently written for Insomnia Studyroom.
Official curriculum reference: College Board AP Physics 1 course page. The explanation and worked example are independently written for this study site.