Directions:Work each problem on your own paper first. For multiple choice, select A, B, C, or D. You will see right away whether you are correct. If you miss, you will see the most likely reason, and you can try again. For short response, type your answer, then compare it with the model solution. Questions mix the five essential knowledge threads of Topic 3.3 — what potential energy is and the freedom to choose U = 0, the force–energy relationship, reading equilibrium from a U(x) graph, the potential energy of common systems, and the total potential energy of multi-object systems. Use g = 10 m/s² unless a problem gives you other values to use. Round decimals to the nearest hundredth unless noted otherwise.
Which of these two-object systems has a well-defined potential energy? System I: a ball and the Earth, interacting only through gravity. System II: a hockey puck and the ice it's sliding across, interacting only through kinetic friction.
Select A, B, C, or D to check your answer.
A 5 kg object sits on a table 0.8 m above the floor, and the floor is 2.2 m above the basement level. (a) Find the object's gravitational PE taking the floor as U = 0. (b) Find the object's gravitational PE taking the basement level as U = 0. (c) The object falls from the table down to the floor. Find ΔU using each choice of zero, and confirm they agree.
Which statement correctly describes potential energy?
Select A, B, C, or D to check your answer.
A 2 kg block is pushed across a rough horizontal floor from point P to point Q along two different routes: a straight 3 m path, and a longer, winding 5 m path, with a constant 8 N kinetic friction force acting along either route. (a) Find the energy dissipated by friction along each route. (b) Explain why these two different values mean friction cannot have an associated potential energy function. (c) Contrast this with how much work gravity would do on the same block if it were instead lifted straight up 3 m along one route, versus up a longer, slanted ramp that still rises a net 3 m along another route.
A system's potential energy is U(x) = 20x² (in joules, x in meters). What is the force on the system at x = 0.5 m?
Select A, B, C, or D to check your answer.
An object experiences a constant conservative force of 15 N in the +x direction as it moves from x = 0 to x = 6 m. (a) Use ΔU = −∫F·dr to find the system's change in potential energy. (b) Does potential energy increase or decrease as the object moves in the direction of the force? (c) Explain this result using the idea that a conservative force points toward decreasing potential energy.
A conservative force points in the +y direction at a certain location. In which direction does the system's potential energy increase fastest from that location?
Select A, B, C, or D to check your answer.
A system has potential energy U(x) = 2x³ − 18x (in joules, x in meters). (a) Find F(x) using F(x) = −dU/dx. (b) Evaluate the force at x = 1 m and state its direction. (c) Evaluate the force at x = 4 m and state its direction.
A graph of U(x) has a local minimum at x = 2 m. What happens to a system displaced slightly to x = 2.3 m?
Select A, B, C, or D to check your answer.
A system's potential energy is U(x) = x² − 4x + 7 (in joules, x in meters). (a) Find the x-value where dU/dx = 0. (b) Find F(x) at x = 1 m and at x = 3 m. (c) Use your results from (b) to determine whether the equilibrium you found in (a) is stable or unstable.
Which of the following always occurs at a local maximum of a system's potential energy function U(x)?
Select A, B, C, or D to check your answer.
The graph shows a system's potential energy U(x), which has a local maximum at x = −2 m and a local minimum at x = 3 m. (a) Classify each equilibrium as stable or unstable. (b) If the system is released from rest at x = −1.5 m (just to the right of the local maximum), which equilibrium will it tend toward? (c) Explain your reasoning using the slope of U(x) at x = −1.5 m.
A spring has spring constant k = 400 N/m. How much elastic potential energy is stored in the spring when it is compressed 0.05 m from its natural length?
Select A, B, C, or D to check your answer.
An elevator carries a 70 kg person up 15 m inside a building, along a path that also shifts 2 m sideways due to the shaft's slight angle. (a) Find the change in the person's gravitational potential energy using the near-surface approximation. (b) Explain why the 2 m sideways shift doesn't affect your answer. (c) Would your approach change if the building were instead 4000 km tall (comparable to Earth's radius)? Explain.
Which of the following situations requires the general gravitational potential energy formula, Ug = −Gm1m2/r, rather than the near-surface approximation ΔUg = mgΔy?
Select A, B, C, or D to check your answer.
A 1000 kg spacecraft moves away from a planet of mass 4.0×10²³ kg (G = 6.67×10−¹¹ N·m²/kg²). (a) Find the planet–spacecraft system's gravitational PE at r = 5.0×10⁶ m. (b) Find it again at r = 2.0×10⁷ m. (c) Find ΔU, and state whether the spacecraft's potential energy increased or decreased as it moved away.
A system contains three objects. Pairwise potential energies are: U₁₂ = −12 J, U₁₃ = +5 J, and U₂₃ = −8 J. What is the system's total potential energy?
Select A, B, C, or D to check your answer.
Three point masses lie on a line: m₁ = 4 kg at x = 0 m, m₂ = 6 kg at x = 5 m, and m₃ = 10 kg at x = 12 m. Using Ug = −Gmamb/r for each pair (G = 6.67×10−¹¹ N·m²/kg²), find the system's total gravitational potential energy.
| Pair | masses (kg) | separation r (m) |
| 1–2 | 4, 6 | 5 |
| 1–3 | 4, 10 | 12 |
| 2–3 | 6, 10 | 7 |
A system consists of a 2 kg block hanging from a spring (k = 150 N/m, stretched 0.2 m) at a height of 1.5 m above the floor. Which of the following correctly identifies the pairwise potential energies that must be summed to find this system's total potential energy, assuming only the block, spring, and Earth interact?
Select A, B, C, or D to check your answer.
A 4 kg block rests against a spring (k = 120 N/m, compressed 0.3 m from its natural length) at the top of a frictionless ramp, 2.0 m above the ground. (a) Find the spring's elastic potential energy. (b) Find the block's gravitational potential energy relative to the ground. (c) Find the system's total potential energy relative to the ground. (d) If the spring's compression were doubled (to 0.6 m) while the height stayed the same, would the system's total potential energy also double? Justify your answer numerically.