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3.2.5Work, Energy, and Power

Gravitational Potential Energy Between Objects — AP PHYSICS 1 Revision Notes & Topical Questions

5 min read 9 parts Lesson 8 of 14 1 quiz

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Lesson summary

3.2.5 Gravitational Potential Energy Between Objects — AP PHYSICS 1 notes: Use UG = −Gm1m2/r for two masses at any separation, including orbital and escape problems.Limit of mgΔy — it works only near a planet's surface where g is nearly constant.Far from a planet — the gravitatio…

  • Resemblance to Fg
  • Zero level
  • Negative values
  • Closer objects
  • Bound system
  • Sum over pairs

Section 1 of 9Why a new formula

Limit of mgΔy
  • Limit of mgΔy it works only near a planet's surface where g is nearly constant.
Far from a planet
  • Far from a planet the gravitational field strength is no longer constant, so a general expression is needed.
UG
−Gm1m2/r
  • absolute gravitational potential energy of two approximately spherical masses.
  • G = 6.67 × 10−11 N·m2/kg2.
  • m1 and m2 are the masses in kg.
  • r is the centre-to-centre separation in m.
Resemblance to Fg
  • Resemblance to Fg the form is like Fg = Gm1m2/r2, with Fg multiplied by r for the potential energy.

Section 2 of 9Meaning of the minus sign

Zero level
  • Zero level potential energy is zero at infinite separation, where the field strength tends to zero.
Negative values
  • Negative values attraction means bringing masses together releases energy, so UG is always negative.
Closer objects
  • Closer objects the smaller r is, the more negative UG becomes.
Bound system
  • Bound system energy must be added to separate the masses to infinity.
Graph
  • UG against r starts very negative and rises toward zero from below, never positive.
  • The gradient magnitude at any r is Gm1m2/r2, the gravitational force.

Section 3 of 9Several objects

Several objects
Scalar
  • Scalar gravitational potential energy is a scalar.
Sum over pairs
  • Sum over pairs for more than two objects the total is the sum over each pair.
Central body with two satellites
  • Ug,sys = Ug1 + Ug2.

Section 4 of 9Changes and conservation

Changes and conservation
Change from r1 to r2
  • ΔUG = −GMm(1/r2 − 1/r1).
  • Moving away (r2 > r1) gives a positive change.
Agreement near the surface
  • Agreement near the surface for small Δy it reduces to mgΔy, because −GMm/(R + h) + GMm/R ≈ GMmh/R2 = mgh.
Conservation
  • Conservation with only gravity doing work, ½mv12 − GMm/r1 = ½mv22 − GMm/r2.

Section 5 of 9Escape speed

Condition
  • Condition the object reaches r → ∞ with speed that can be zero, so the total energy at launch is zero.
Equation
  • ½mvesc2 − GMm/R = 0, so vesc = √(2GM/R).
Earth
  • Earth about 11.2 km/s, independent of the object's mass and ignoring air resistance.
Total energy sign
  • Total energy sign negative means bound; zero or positive means it escapes.
    • (returns or orbits)

Section 6 of 9Energy in circular orbits

Kinetic energy
  • GMm/r2 = mv2/r gives K = GMm/(2r).
Total energy
  • E = K + UG = −GMm/(2r), negative with magnitude equal to K.
Potential energy
  • UG = −GMm/r has twice the magnitude of K.
Higher orbit
  • Higher orbit total energy is less negative though speed decreases, so raising an orbit needs positive work.
Changing orbits
  • ΔE = (GMm/2)(1/r1 − 1/r2).

Section 7 of 9More from the specification

1Gravitational binding and reference frames
  • Gravitational binding and reference frames radial distance between interacting celestial masses defines mutual gravitational potential, independent of external observer reference frames.
2Observer
  • Person or instrument measuring from within a frame.
3Driver
  • The external system supplying the periodic force.
4Scalar quantity
  • A quantity with magnitude only, such as potential
5Reference frame
  • Coordinate system used by an observer.
6Spring constant
  • Stiffness k in N/m.
7Elastic
  • Returns to original shape when force is removed.
8Velocity
  • Rate of change of displacement.
9Distance
  • Total length of path travelled, a scalar.
10Stationary
  • Not moving.

Section 8 of 9Key terms

TermMeaning
Gravitational potential energyUG = −Gm1m2/r for two masses.
Bound systemSystem with negative total energy.
Escape speedMinimum launch speed to reach infinity, √(2GM/R).
InfinityReference separation where UG = 0.
Total energyK + U.

Section 9 of 9Exam technique

1
Dropping the minus sign and treating UG as positive.
2
Using surface-to-surface distance for r.
3
Using mgΔy for large height changes.
4
Forgetting that the zero of UG is at infinity, not at the surface.
5
Thinking escape speed depends on the object's mass.

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