Conservative Forces and Gravitational Potential Energy

Core idea

Conservative work depends on endpoints

A conservative force does the same work between two positions no matter which path connects them. Gravity is conservative near Earth, so a direct drop and a longer frictionless ramp produce the same gravitational work when they have the same vertical change.

W_c = −ΔU
U_g = mgy    and    ΔU_g = mgΔy

Gravitational potential energy belongs to the object–Earth system. The height-zero line is a reference choice, not a physical surface. Moving that line shifts the numerical U_g values but leaves energy differences, forces, and motion unchanged.

Guided lesson path

Compare paths, references, and energy accounting

1. Observe a non-conservative force. A 2 kg block sliding against 4 N of kinetic friction loses mechanical energy according to the distance traveled.
2. Measure friction on a 4 m path. Friction points opposite displacement, so W_f = (4 N)(4 m) cos180° = −16 J.
3. Measure friction on a 6 m path. The same friction force now does −24 J. The endpoints alone do not determine non-conservative work.
4. State the path-dependence test. If two routes between the same positions produce different work, the force is non-conservative in the model.
5. Test a conservative force. A 2 kg block descends 4 m directly. Gravity does W_g = mgΔh = (2)(9.8)(4) = +78.4 J.
6. Take a longer frictionless route. The same 4 m vertical drop along an incline also produces +78.4 J of gravitational work. Ramp length does not change the result.
7. Compare gravitational paths. Equal endpoints give equal gravitational work and equal kinetic-energy changes when other work is absent.
8. Complete a closed path. A block that rises and returns to its starting height has zero net gravitational work because its initial and final positions are the same.
9. Classify common forces. Gravity and ideal spring forces are conservative. Kinetic friction, air resistance, and a person’s push are generally treated as non-conservative transfers.
10. Assign potential energy to a system. U_g describes the configuration of the object–Earth system, not energy stored inside the object alone.
11. Choose a reference level. Set U_g = 0 at a convenient height, then measure signed height from that line. The reference changes bookkeeping, not motion.
12. Calculate a baseline value. A 2 kg block 3 m above the reference has U_g = (2)(9.8)(3) = 58.8 J.
13. Compare two heights. Raising the same 2 kg block from 3 m to 6 m doubles U_g from 58.8 J to 117.6 J because U_g is proportional to height.
14. Compare two masses. At 3 m, a 6 kg block has U_g = 176.4 J, three times the 2 kg value. Gravitational potential energy is proportional to mass at fixed height.
15. Read the ground reference. A 2 kg block at y = 5 m with y₀ = 0 has U_g = 98.0 J.
16. Shift the zero upward. If y₀ changes to 2 m, the same block is 3 m above the reference and U_g becomes 58.8 J. Its motion has not changed.
17. Compare the reference choices. Individual U_g values shift, but ΔU_g between the same two positions and the predicted motion remain unchanged.
18. Relate gravitational work and potential energy. During a 4 m downward displacement, ΔU_g = −78.4 J while W_g = +78.4 J. The signs are opposite.
19. Compare conservative and non-conservative accounting. Gravity transfers energy between U_g and K without path dependence; friction removes mechanical energy in proportion to the actual path.
20. Finish with the endpoint test. Name the system, identify the force, compare paths, select a reference, calculate ΔU, and then use W_c = −ΔU or the appropriate work sum.

Worked examples

Keep the endpoint change and sign together

For a 2 kg block falling 4 m:

ΔU_g = mgΔy = (2 kg)(9.8 m/s²)(−4 m) = −78.4 J
W_g = −ΔU_g = +78.4 J

For the same block descending a longer frictionless ramp:

Δy = −4 m, regardless of ramp length
W_g = (2 kg)(9.8 m/s²)(4 m) = +78.4 J

For a 2 kg block 5 m above a reference at y₀ = 2 m:

h = y − y₀ = 5 m − 2 m = 3 m
U_g = mgh = (2 kg)(9.8 m/s²)(3 m) = 58.8 J

For 4 N kinetic friction over 6 m:

W_f = f_kd cos180° = (4 N)(6 m)(−1) = −24 J

Path test

Same endpoints do not always mean same work

Gravity is conservative because only the vertical endpoints matter in a uniform gravitational field. A longer ramp can change the force component along the surface and the travel distance, but those changes compensate so the total gravitational work remains mg times the vertical drop.

Friction behaves differently. A longer route creates more contact distance, so the same kinetic-friction force transfers more energy out of the moving object. If the work changes when the route changes, do not represent that force with one endpoint-only potential-energy value.

Reference levels

Changing zero changes values, not predictions

Potential energy is defined relative to a chosen zero. With U_g = mg(y − y₀), changing y₀ adds or subtracts the same constant from every position. The measurable difference between two positions is unchanged:

ΔU_g = mg[(y_f − y₀) − (y_i − y₀)] = mg(y_f − y_i)

Choose the ground, the starting point, or another convenient level. State the choice so another reader can interpret the numerical values, then use differences for physical predictions.

System accounting

Potential energy belongs to the interacting system

Gravitational potential energy describes the object–Earth configuration. If Earth is outside the chosen system, gravity can appear as external work. If the object and Earth are both included, gravity is an internal conservative interaction and its energy can be tracked as U_g.

This boundary choice changes the bookkeeping language, not the physical result. Be explicit about which objects are in the system before deciding whether energy is stored internally or transferred by work across the boundary.

For teachers

Make path and reference choices explicit

Start with the 2 kg, 4 m gravitational drop and have students predict the work before running the longer ramp. Then use the 4 m and 6 m friction trials to make path dependence visible with the same force magnitude.

Have students record U_g at y₀ = 0 m and then at y₀ = 2 m. Ask them to identify which values shift and which differences remain fixed. Finish by requiring a system boundary before students label gravity as internal potential-energy storage or external work.

Continue with Work and Kinetic Energy for force–displacement work, Conservation of Mechanical Energy for energy transformations, and Conservation of Energy with External Work for system-boundary accounting.