AP® Physics 1: Algebra-Based review sheet from Aim for Five (aimforfive.com/physics/units/3/3-3)
Unit 3 · Topic 3.3
3.3 Potential Energy
Potential energy is energy stored in the arrangement of a system of objects that interact through conservative forces, such as gravity or a spring. Near Earth, ΔU_g = mgΔy; a spring stores U_s = ½kx²; and for planets and stars, U_G = −Gm₁m₂/r. Only changes in potential energy have physical meaning.
Key terms
- gravitational potential energy
- elastic (spring) potential energy
- conservative force
- zero of potential energy
- system
Potential energy belongs to a system
Potential energy is stored in the configuration (the arrangement) of a system of two or more objects that interact through conservative forces. A raised book doesn't "have" gravitational potential energy by itself; the book–Earth system does, because of their separation. A single object treated as a point can't store potential energy.
Potential energy is a scalar that depends on the positions of the objects in the system. Each conservative force has its own kind: gravitational potential energy for gravity and elastic potential energy for springs. Friction has none, because its work depends on the path.
Gravitational potential energy near Earth
Near Earth's surface, g is nearly constant, so the change in gravitational potential energy when an object of mass m moves vertically by Δy is ΔU_g = mgΔy. Raise it and U_g increases; lower it and U_g decreases. Horizontal motion doesn't change U_g, and the path between two heights doesn't matter.
You choose where U_g = 0. The floor, the tabletop or the lowest point of a swing all work. Changing that choice changes every value of U_g by the same amount, so the changes, which are what matter, stay the same. With the zero at a high point, U_g below it is negative, and that's fine.
Elastic potential energy
An ideal spring stretched or compressed a distance x from its relaxed length stores U_s = ½kx² (written ½k(Δx)² on the equation sheet). Because x is squared, U_s is never negative, and stretching or compressing by the same amount stores the same energy. Doubling x quadruples the stored energy.
For springs the zero is not a free choice in the formula: U_s = ½kx² puts zero at the relaxed length. This energy equals the work you do to stretch the spring slowly, the triangle-shaped area under the force–stretch graph.
Gravitational potential energy in space
When distances are large, as with planets, moons, stars and satellites, g isn't constant. For two roughly spherical masses with centers a distance r apart, U_G = −Gm₁m₂/r.
This formula sets U_G = 0 when the objects are infinitely far apart. Every closer arrangement has less energy, so U_G is always negative. Moving the objects apart makes U_G less negative, which is an increase. mgΔy is just this formula's approximation for small height changes near a surface.
Systems of several objects and graphs
If a system has more than two objects, its total potential energy is the sum over every pair. For three masses, that's three pair terms: 1–2, 1–3 and 2–3.
Graph shapes worth knowing: U_g against height near Earth is a straight line with slope mg. U_s against x is an upward parabola with its lowest point at x = 0. U_G against r is a curve that starts very negative at small r and rises toward zero as r grows, never crossing it.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Raising a book and choosing the zero
A 0.50 kg book is lifted from the floor to a shelf 1.8 m higher. Use g = 9.8 m/s². Find the change in the book–Earth system's gravitational potential energy. Then find U_g at the floor and at the shelf if (a) the floor is chosen as zero, and (b) the shelf is chosen as zero.
Show the solutionHide the solution
- Step 1: ΔU_g = mgΔy = (0.50)(9.8)(1.8) ≈ 8.8 J.
- Step 2: (a) Floor = 0: U_floor = 0 and U_shelf = +8.8 J.
- Step 3: (b) Shelf = 0: U_shelf = 0 and U_floor = −8.8 J.
- Step 4: Either way ΔU_g = +8.8 J. The choice of zero changes the values, not the change.
Answer: ΔU_g ≈ +8.8 J in both cases; (a) 0 J and 8.8 J; (b) −8.8 J and 0 J
- Example 2Calculator allowed
Spring energy and compression (classic trap)
A spring with k = 400 N/m is compressed 5.0 cm. How much energy does it store? How much if it's compressed 10.0 cm instead?
Show the solutionHide the solution
- Step 1: Convert: 5.0 cm = 0.050 m. U_s = ½kx² = ½(400)(0.050)² = 0.50 J.
- Step 2: At 0.100 m: U_s = ½(400)(0.100)² = 2.0 J.
- Step 3: Doubling the compression gave 4 times the energy, because x is squared. The trap is expecting 1.0 J.
Answer: 0.50 J; 2.0 J
- Example 3Calculator allowed
Raising a satellite's orbit
A 1000 kg satellite moves from an orbit of radius 7.0 × 10⁶ m to one of radius 1.4 × 10⁷ m around Earth (mass 5.97 × 10²⁴ kg). Find the satellite–Earth system's gravitational potential energy in each orbit and the change.
Show the solutionHide the solution
- Step 1: U_G = −GMm/r. First orbit: −(6.67 × 10⁻¹¹)(5.97 × 10²⁴)(1000) ÷ (7.0 × 10⁶) ≈ −5.69 × 10¹⁰ J.
- Step 2: Second orbit: r doubled, so U_G is half as negative: ≈ −2.84 × 10¹⁰ J.
- Step 3: ΔU_G = (−2.84 × 10¹⁰) − (−5.69 × 10¹⁰) ≈ +2.84 × 10¹⁰ J. Moving apart increased the potential energy, even though both values are negative.
- Step 4: You can't use mgΔy here: the change in height is about 7000 km, and g changes a lot over that distance.
Answer: About −5.7 × 10¹⁰ J and −2.8 × 10¹⁰ J; change ≈ +2.8 × 10¹⁰ J
Common mistakes
- Saying a single object "has" potential energy. Potential energy belongs to a system of interacting objects, such as object plus Earth.
- Thinking a negative U_g or U_G is impossible. Negative values just mean below your chosen zero; changes are what matter.
- Using the spring's total length for x in ½kx². Use the stretch or compression from the relaxed length.
- Using mgΔy for large changes in distance from a planet. Use −Gm₁m₂/r when g changes noticeably.
On the exam
- Expect questions that ask which system has potential energy, or how U changes when a distance doubles. Doubling a spring's stretch makes U_s 4 times as large; doubling the distance between two planets halves U_G = −Gm₁m₂/r (it becomes less negative).
- On free response, state your zero of gravitational potential energy and keep it consistent across all your energy equations and bar charts.
Connected topics
Videos
Check yourself
4 questions on 3.3 Potential Energy. Pick an answer to see if you got it, and why.
A 3.0 kg backpack is lifted from the floor onto a shelf 2.0 m higher. By how much does the gravitational potential energy of the backpack–Earth system change? Use g = 10 m/s².
A spring with k = 400 N/m is compressed 0.10 m from its relaxed length. How much elastic potential energy does it store?
A spring is compressed by x and stores energy U. If it is compressed by 2x instead, how much energy does it store?
Two students analyze a ball falling from a shelf to the floor. One sets U_g = 0 at the floor; the other sets U_g = 0 at the shelf. Which quantity will they agree on?
0 of 4 answered