Potential Energy
Potential energy is a scalar property of a multi-object system’s configuration when the relevant interactions are conservative. Physics C connects it to work through ΔU=−∫F⃗_c·dr⃗ and to force through F_x=−dU/dx. A U(x) graph reveals force and equilibrium directly; when a total mechanical-energy level is also known, it can reveal allowed regions, kinetic-energy budget, and turning points.
Topic sequence · 3 of 5
Learning objectives
Objective 1
Describe potential energy as a scalar property of a system configuration and explain why the reference zero can be chosen by the observer.
Objective 2
Calculate changes in potential energy from conservative-force work and infer one-dimensional conservative force from the slope of U(x).
Objective 3
Interpret potential-energy graphs to identify stable and unstable equilibrium, compare spring and gravitational models, and sum pairwise potential energies in multi-object systems.
- Topic 3.2 work, conservative versus nonconservative forces, and the signed work integral.
- Topic 2.8 ideal spring force F_x=−kx.
- Topic 2.6 universal gravitation and nearly uniform gravitational fields.
- Topic 1.2 derivatives and definite integrals.
Definitions first
Representations you must read and create
Representation 1
U-versus-x graph: slope gives −F_x; minima and maxima reveal stable and unstable equilibrium, respectively.
Representation 2
Force-versus-position and potential-versus-position paired graphs showing that integrating F gives −ΔU while differentiating U gives −F.
Representation 3
System diagram identifying the two or more interacting objects whose configuration owns the potential energy; potential energy does not belong to an isolated single object by itself.
Representation 4
Energy reference diagram showing that shifting U(x) vertically by a constant changes the numerical zero but not forces, equilibrium, or energy differences.
Representation 5
Potential-energy landscape with a horizontal total-mechanical-energy line E: allowed positions satisfy E≥U, the vertical gap E−U is K, and intersections E=U are turning points.
Core relationships and calculus
Variables: U potential energy (J); F⃗_c conservative force (N); dr⃗ displacement (m)
Use: calculate a change in potential energy from conservative-force work
Variables: F_x force component (N); U(x) joules; x metres
Use: infer one-dimensional conservative force from the slope of a potential-energy function
Variables: k spring constant (N/m); Δx extension/compression from relaxed length (m)
Use: elastic potential energy of an ideal spring
Variables: G gravitational constant; m₁,m₂ masses (kg); r center-to-center separation (m)
Use: general gravitational potential energy for approximately spherical masses with U→0 at infinity
Variables: m mass (kg); g nearly uniform field strength (N/kg); Δy vertical change (m)
Use: near-surface gravitational potential-energy change when g is approximately constant
Variables: pairwise potential energies (J)
Use: total potential energy of a system containing more than two mutually interacting objects
Potential energy is defined by a conservative-force integral. In one dimension, ΔU=−∫F_x dx. Differentiation reverses that operation: F_x=−dU/dx. The negative sign means the force points toward decreasing U. A constant added to U disappears under differentiation, which is why the zero of potential energy can be chosen for convenience without changing the dynamics.
For a conservative force, define the potential-energy change as the negative of the work done by that force: ΔU=−∫F⃗_c·dr⃗. In one dimension, take an infinitesimal displacement dx: dU=−F_x dx. Dividing by dx gives dU/dx=−F_x, or F_x=−dU/dx. For an ideal spring F_x=−kx, integrating from 0 to x gives U_s(x)−U_s(0)=−∫₀ˣ(−kx′)dx′=½kx². For a gravitational force F_r=−GMm/r², integrating with U(∞)=0 gives U_g=−GMm/r.
Potential energy is associated with conservative interactions and is defined only up to an arbitrary additive constant. A U(x) graph alone locates equilibrium from slope; turning points require a specified total-energy level E.
Physical examples
Example 1
Compressing or stretching an ideal spring by the same magnitude stores the same U_s=½kx² even though the spring forces point in opposite directions on the two sides of equilibrium.
Example 2
For a satellite–planet system, U_g=−GMm/r becomes more negative as separation decreases. Choosing U=0 at infinity makes a bound gravitational configuration naturally have negative potential energy.
Example 3
Near Earth’s surface, lifting a mass through height Δy increases gravitational potential energy by approximately mgΔy because the gravitational field changes negligibly over that displacement.
Example 4
On a U(x) graph, a local minimum marks stable equilibrium while a local maximum marks unstable equilibrium; the force at any point points “downhill” because F_x=−dU/dx.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
On a U(x) graph, find force from the negative slope before making a motion claim. A horizontal tangent means F_x=0, but the curvature or nearby slope pattern determines stability.
Reasoning pattern 2
When a problem allows a convenient zero, choose it to simplify algebra, but keep the same reference throughout the calculation. Only differences in U should affect observable motion.
Reasoning pattern 3
Choose the correct gravitational model: mgΔy for a nearly uniform field over a small height change, −GMm/r for large-scale spherical gravity.
Reasoning pattern 4
For multiple objects, define the system first and count each conservative interacting pair once when building U_total.
Reasoning pattern 5
Do not identify turning points from U(x) alone. A turning point requires K=0, so compare U(x) with the stated total mechanical energy E and solve E=U(x).
Deeper calculus and limiting-case check
Near a stable minimum x₀, many smooth potential-energy functions can be approximated by a parabola: U(x)≈U(x₀)+½U″(x₀)(x−x₀)². Then F≈−U″(x₀)(x−x₀), which has the same mathematical form as a spring restoring force. This local approximation foreshadows why many small oscillations behave approximately as simple harmonic motion in Unit 7.
Model-selection workflow: decide what the problem is before choosing an equation
For Potential Energy, the fastest reliable route is not equation hunting. Start by identifying the system boundary, encode it with the energy/work representation, choose the work-energy or conservation equation, and then test the conservative-force and transfer assumption. This sequence prevents a mathematically correct manipulation from being attached to the wrong physical model.
Define the physical object or system
Name what is being modeled, what is outside the system, the coordinate or orientation convention, and the interval or geometry over which the statement applies. Treat potential energy and conservative interaction as physical quantities first and symbols second.
Choose a representation
Build U-versus-x graph: slope gives −F_x; minima and maxima reveal stable and unstable equilibrium, respectively.. If the prompt supplies Force-versus-position and potential-versus-position paired graphs showing that integrating F gives −ΔU while differentiating U gives −F., translate it into the same sign, axis, orientation, or reference convention before combining information.
Write the governing relationship
A useful starting relationship is ΔU = −W_c = −∫ₐᵇ F⃗_c·dr⃗. State what the variables mean and why this relation—not merely a familiar formula—matches the physical situation.
Check conditions and limits
Potential energy is associated with conservative interactions and is defined only up to an arbitrary additive constant. A U(x) graph alone locates equilibrium from slope; turning points require a specified total-energy level E. After solving, test units, sign or direction, zero/large-parameter limits, and whether the answer remains consistent with the original representation.
Representation → equation → calculus → physics chain
| Stage | What to write or inspect | What proves the step is physically valid |
|---|---|---|
| Representation | U-versus-x graph: slope gives −F_x; minima and maxima reveal stable and unstable equilibrium, respectively. | Every arrow, sign, axis, node, surface, path, or reference has an explicit meaning. The diagram must contain enough information to determine the relevant components or constraints. |
| Primary model | ΔU = −W_c = −∫ₐᵇ F⃗_c·dr⃗ | calculate a change in potential energy from conservative-force work. Preserve the variables and units long enough to see dependence before inserting numerical values. |
| Second relationship | F_x = −dU/dx | infer one-dimensional conservative force from the slope of a potential-energy function. Combine relationships only after confirming they describe the same system, interval, frame, or geometry. |
| Calculus operation | Potential energy is defined by a conservative-force integral. In one dimension, ΔU=−∫F_x dx. Differentiation reverses that operation: F_x=−dU/dx. The negative sign means the force points toward decreasing U. A constant added to U disappears under differentiation, which is why the zero of potential energy can be chosen for convenience without changing the dynamics. | State what the derivative, integral, differential equation, or limiting process represents physically; do not treat calculus notation as an algebraic decoration. |
| Physical check | Return to the original words and representation. | The final sign, direction, units, endpoint behavior, and functional dependence must all agree. A correct-looking expression that violates one of these checks needs repair. |
Advanced synthesis: how to handle an unfamiliar version of this topic
Suppose a one-dimensional system has U(x)=3x²−12x+7 J, with x in metres. Then F_x=−dU/dx=−(6x−12)=12−6x N. Equilibrium occurs where F_x=0, so x=2 m. Because d²U/dx²=6 J/m²>0, the equilibrium is stable and the graph has a local minimum there. At x=0, U=7 J and F_x=+12 N; at x=3 m, U=−2 J and F_x=−6 N. The additive constant in U shifts every energy value but would not change F_x or the equilibrium location. To deepen this example, change one assumption or parameter, predict the effect before recomputing, and then use the symbolic dependence to verify the prediction.
FRQ-quality response architecture for Topic 3.3
Mathematical routines
Start from the governing law or definition, keep the derivation symbolic, identify the condition that permits any simplification, and finish with units plus a limiting-case check. For this topic, the relationship ΔU = −W_c = −∫ₐᵇ F⃗_c·dr⃗ should appear only when its physical conditions have been established.
Translation between representations
Move deliberately between U-versus-x graph: slope gives −F_x; minima and maxima reveal stable and unstable equilibrium, respectively. and Force-versus-position and potential-versus-position paired graphs showing that integrating F gives −ΔU while differentiating U gives −F.. Explain what a slope, area, direction, curvature, intercept, node value, field orientation, or other feature means instead of merely redrawing it.
Experimental design and analysis
Turn the topic into a measurable relationship. State the independent and dependent variables, important controls, repeated/ranged trials, and the graph or transformed graph whose slope/intercept would test the model or extract a physical parameter.
Qualitative/quantitative translation
Make the comparison or direction claim first, justify it through the model, derive the symbolic dependence independently, and then reconcile the derivation with the original qualitative claim.
Do not let an equation stand in for an explanation. A complete response names what changes, identifies the governing physical relationship, follows the causal chain, and states the requested outcome explicitly.
Experiment and data-analysis upgrade
Use a force sensor to stretch an ideal spring quasi-statically through a measured displacement. Plot force versus displacement, numerically integrate the force curve to obtain stored elastic energy, and compare the result with ½kx² using a separately measured slope k. Then differentiate or estimate the slope of a fitted U(x) curve to reconstruct the spring force, discussing force-sensor zero and displacement uncertainty.
Operationalize the variables
Say exactly what instrument or observable supplies each measured quantity. Distinguish a directly measured variable from a quantity calculated later from those measurements.
Predict the graph before collecting data
Use the governing relationship to decide what should be plotted. If the raw relationship is nonlinear, choose a transformation that produces a line whose slope or intercept has a defined physical meaning.
Control and repeat for a reason
Hold variables fixed when they would change the predicted relationship. Repeat trials to estimate scatter; vary the independent quantity over a useful range to test functional dependence. These solve different experimental problems.
Evaluate the model, not only the number
Compare the best-fit result with the theoretical prediction, inspect residual or endpoint behavior when appropriate, and identify whether a discrepancy is random scatter, instrument resolution, or a systematic violation of an ideal-model assumption.
Error-repair matrix
| Failure pattern | Repair move | Self-check |
|---|---|---|
| Treating the zero of potential energy as an absolute physical requirement. The reference level is chosen for convenience; physically meaningful predictions depend on ΔU or derivatives of U. | Return to the energy/work representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Dropping the negative sign in F_x=−dU/dx. A positive slope of U means the conservative force points in the negative x-direction. | Return to the work-energy or conservation equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Calling every equilibrium stable. Local minima of U are stable; local maxima are unstable. A flat region requires additional analysis rather than a memorized label. | Return to the energy/work representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Using mgh for large changes in distance from a planet where g is not approximately constant. Use U_g=−GMm/r for the general spherical-gravity model. | Return to the work-energy or conservation equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Adding gravitational or electric-style pair energies incorrectly for many-body systems. Count each interacting pair once when forming the total potential energy. | Return to the energy/work representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
Spiral review: make this topic transferable
A topic is exam-ready only when it can be recognized inside a mixed problem. Use the connections below as transfer tests rather than as optional reading.
Transfer 1
Topic 3.2 defines conservative work and provides the integral that becomes ΔU=−W_c. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 2
Topic 2.8 supplies the spring force whose integral produces U_s=½kx². Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 3
Topic 2.6 supplies the gravitational force whose integral produces U_g=−GMm/r and the near-surface mgΔy approximation. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Formula-condition retrieval
From memory, write ΔU = −W_c = −∫ₐᵇ F⃗_c·dr⃗ and one condition that must be true before using it. Then state one situation where a more general derivative, integral, conservation law, or field/force model is required.
Topic 3.3 mastery ladder
Interactive potential-energy curve analyzer
For U(x)=ax²+bx+c, inspect U, force Fₓ=−dU/dx, the equilibrium point, and stability. Changing c shifts the energy reference without changing the force.
Experimental connection
Use a force sensor to stretch an ideal spring quasi-statically through a measured displacement. Plot force versus displacement, numerically integrate the force curve to obtain stored elastic energy, and compare the result with ½kx² using a separately measured slope k. Then differentiate or estimate the slope of a fitted U(x) curve to reconstruct the spring force, discussing force-sensor zero and displacement uncertainty.
Connections across Physics C
Connection 1
Topic 3.2 defines conservative work and provides the integral that becomes ΔU=−W_c.
Connection 2
Topic 2.8 supplies the spring force whose integral produces U_s=½kx².
Connection 3
Topic 2.6 supplies the gravitational force whose integral produces U_g=−GMm/r and the near-surface mgΔy approximation.
Connection 4
Topic 3.4 combines K and U into mechanical-energy accounting and uses system boundaries to decide when that quantity is conserved.
Related lessons and resources
Use these links when the current model depends on a prerequisite or becomes the foundation for the next model.
Common mistakes
Avoid this
Treating the zero of potential energy as an absolute physical requirement. The reference level is chosen for convenience; physically meaningful predictions depend on ΔU or derivatives of U.
Avoid this
Dropping the negative sign in F_x=−dU/dx. A positive slope of U means the conservative force points in the negative x-direction.
Avoid this
Calling every equilibrium stable. Local minima of U are stable; local maxima are unstable. A flat region requires additional analysis rather than a memorized label.
Avoid this
Using mgh for large changes in distance from a planet where g is not approximately constant. Use U_g=−GMm/r for the general spherical-gravity model.
Avoid this
Adding gravitational or electric-style pair energies incorrectly for many-body systems. Count each interacting pair once when forming the total potential energy.
Mastery check
- Define the main quantity without relying on an equation.
- Choose the correct representation and sign convention.
- Explain why the calculus relationship matches the physics.
- Design or interpret an experiment that could test the model.
Frequently asked questions
Why can potential energy be negative?
The zero of U is a chosen reference. Negative U simply means the chosen configuration lies below that reference; for gravity it is common to choose U=0 at infinite separation, making bound configurations negative.
Does potential energy belong to one object?
In the mechanics model, potential energy is associated with the configuration of a system of interacting objects. Saying “the object has mgh” is shorthand for the object–planet system’s gravitational potential energy.
How do I tell stable from unstable equilibrium on U(x)?
Equilibrium occurs where dU/dx=0. A local minimum is stable because small displacements produce a restoring force; a local maximum is unstable because small displacements produce a force away from the point.
When should I use mgh instead of −GMm/r?
Use mgΔy for changes near a planetary surface when g is approximately constant. Use −GMm/r when the variation of g with distance matters.