Work, Energy and Power · Lesson 5 of 16
Potential Energy and Conservative Forces
“Define stored energy through work and connect force to the slope of potential energy.”
• Define potential-energy change with its sign. • Recognise path independence and closed-loop tests. • Explain the arbitrary zero of potential energy.
Stored energy belongs to a configuration
Raise a stone, stretch a bow or compress a spring: each rearrangement can make motion possible later. Potential energy describes energy stored through configuration. More precisely, gravitational potential energy belongs to the Earth–stone interaction and elastic potential energy to the spring system. Saying “the stone’s potential energy” is convenient shorthand when Earth is treated as fixed. An arbitrary force does not necessarily have a single-valued potential-energy function.
Define the change by conservative work
A conservative force does positive work when its potential energy decreases: stored energy is released. When you raise a stone slowly, gravity does negative work, so gravitational potential energy increases. Your lifting work equals this increase only when the stone’s kinetic energy does not change and other work is absent. A rapid lift can increase both kinetic and potential energy.
Gravity near Earth
Problem
A 2 kg object rises from y = −1 m to y = 3 m. Use g = 10 m/s². Compare reference choices.
- 1.The increase is mgΔy = 2×10×4 = 80 J. Gravity does −80 J of work.
- 2.With U = mgy, Ui = −20 J and Uf = 60 J. Negative potential energy is allowed.
- 3.With zero chosen at y = −1 m, Ui = 0 and Uf = 80 J. The difference remains 80 J.
- 4.Changing the additive constant changes neither forces nor physical predictions. Keep the same reference throughout a calculation.
Three tests for a conservative force
First, work between two points depends only on those endpoints, not on which route connects them. Second, work around every closed route is zero: the return journey exactly cancels the outward conservative work. Third, in one dimension a position-dependent conservative force can be written as the negative derivative of a potential. These statements express the same energy bookkeeping when the field and allowed domain meet the usual conditions. A single loop with zero work is insufficient to prove a force is conservative on every route.
If U rises with x, its slope is positive and force points toward −x. If U falls with x, force points toward +x. A flat region gives zero conservative force. At a smooth minimum the restoring force points back toward the minimum after a small displacement; at a maximum it points away. Equilibrium and stability are developed further with potential-energy graphs.
Path dependence of friction
Problem
A block slides between two points 3 m apart on a horizontal rough surface. Friction magnitude is 4 N. Compare a direct 3 m path with a 5 m detour.
- 1.Direct friction work is −4×3 = −12 J; the detour gives −4×5 = −20 J.
- 2.Returning by another 3 m path adds another −12 J, not +12 J. A 6 m closed slide gives −24 J.
- 3.Since friction’s work depends on the route and does not cancel on a loop, no ordinary position-only potential energy accounts for sliding friction.
A comet on an elliptical orbit is a useful counterexample to “gravity never does work in orbit.” Gravity is usually not perpendicular to the velocity on an ellipse, so its instantaneous work can be nonzero. Over a complete orbit the comet returns to the same configuration and gravity’s net work is zero because it is conservative.
Check your understanding
Quiz
Positive conservative work means U
Explanation: Wc = −ΔU.
Adding a constant to U changes
Explanation: A derivative and a difference remove the constant.
If dU/dx is positive, Fx is
Explanation: Force is minus the slope.
Sliding friction over a closed trip generally does
Explanation: It opposes relative sliding on the outward and return segments.
Gravity’s work in a complete closed orbit is
Explanation: Initial and final gravitational potential energy are equal.
Key Takeaways
• Potential energy is defined through conservative work. • Only differences in U matter; its zero is a choice. • Gravity is conservative; sliding friction needs a different energy account.