Unit 7 · Topic 03
Gravitational potential energy
The reference U(∞)=0 fixes the sign and supports analysis of bound states and escape.
Unit 7Gravitational energyOpen navigation
Concept 01
Negative potential energy
Essential The minimum you should retain
With U(∞)=0, U(r)=−GMm/r is negative at finite distance.
UnderstandInterpret and connect
Separating the masses raises U toward zero; bringing them closer makes it more negative.
DeepenFormulation and conditions
The sign depends on the reference, but differences and force do not.
ExploreConnections for further study
A bound state has negative mechanical energy with this reference.
Mathematical relation
Gravitational potential energy
Potential energy and its radial change.
U is negative at finite distance with this reference.
DeepenVariables, conditions, and checks
Variables
- U
- gravitational potential energy; usual unit: J
Conditions of application
- Reference U(∞)=0; positive radii.
Dimensional check
J.
normalized gravitational potential energy
- U<0
With U(∞)=0, potential energy is negative and approaches zero from below.
Concept 02
Comparing two radii
Essential The minimum you should retain
ΔU=GMm(1/rᵢ−1/r_f) preserves the initial-to-final order.
UnderstandInterpret and connect
If r_f>rᵢ, ΔU>0 and ideal energy input is required.
DeepenFormulation and conditions
The radial force satisfies F_r=−dU/dr=−GMm/r².
ExploreConnections for further study
The slope, rather than the chosen zero, determines the force.
Worked example
Raising a mass from R to 2R
m=1000 kg, μ_E=3.986×10^14 m³/s², and R_E=6.371×10^6 m.
- m=1000 kg
- r_i=R_E
- r_f=2R_E
Find ΔU.
- Model
ΔU=μm(1/r_i−1/r_f).
- Radii
r_i=R and r_f=2R.
- Simplify
ΔU=μm/(2R).
- Result
ΔU≈+3.13×10^10 J.
Concept 03
The local limit
Essential The minimum you should retain
Near a surface and for |Δh|≪R, ΔU≈mgΔh.
UnderstandInterpret and connect
The approximation assumes that g changes little across the region.
DeepenFormulation and conditions
Expanding 1/(R+h) to first order connects the universal expression to mgh.
ExploreConnections for further study
At planetary scales one must return to −GMm/r.
Concept 04
Consistent references
Essential The minimum you should retain
K and U must use the same reference and system.
UnderstandInterpret and connect
Changing the zero adds a common constant to U and E without altering ΔU.
DeepenFormulation and conditions
E=0 is the ideal limit reaching infinity with zero final speed.
ExploreConnections for further study
Losses or propulsion require extra terms in the simple mechanical balance.
Concept review
Common errors
Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.
Using U=mgh at planetary scales.
Use U=−GMm/r; mgh is a local approximation.
Assigning U=+GMm/r when U(∞)=0.
That reference requires U=−GMm/r.