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Theme D — Fields

D.1.7 Gravitational Potential Gradient — IB Physics HL Revision Notes & Topical Questions

5 min read 8 parts Lesson 9 of 39 1 quiz

📚 IB Physics HL Hub / Topical Revision
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Lesson summary

D.1.7 Gravitational Potential Gradient — IB PHYSICS HL notes: Relate gravitational field strength to the negative gradient of the potential and interpret V–r graphs.Definition — the rate of change of gravitational potential with displacement in the direction of the fieldField and…

  • Field and potential
  • Equipotential lines
  • Relation to equation
  • Inward move
  • g = −ΔV/Δr
  • Equation — g = −ΔVg/Δr

Part 1 of 6Potential gradient

Definition
  • Definition the rate of change of gravitational potential with displacement in the direction of the field
Field and potential
  • Field and potential the field at a point is the negative gradient of the potential–distance graph
Equation
  • Equation g = −ΔVg/Δr
Symbols
  • Symbols g in N kg−1, ΔVg in J kg−1 and r from the centre of a point mass
Equipotential lines
  • Equipotential lines they are perpendicular to field lines and show the potential gradient

Part 2 of 6The V–r graph for a planet

Sign
  • Sign all values of Vg are negative, below the r-axis
Shape
  • Vg follows a −1/r relation
Slope
  • Slope a shallow increase as r grows, tending to zero at infinity
Equation
  • Equation it represents Vg = −GM/r with G and M constant

Part 3 of 6Finding g from the graph

Gradient
  • Gradient at any point the gradient equals the field strength g
Tangent
  • Tangent draw a tangent at the point and calculate its gradient
Sign
  • Sign the gradient of V against r is positive, so g = −gradient is negative, which means g points towards the mass
Relation to equation
  • Relation to equation differentiating −GM/r gives GM/r2

Part 4 of 6Potential difference between two radii

Equation
  • ΔVg = −GM(1/r2 − 1/r1)
Inward move
  • Inward move to a smaller radius ΔVg is negative
Method
  • Method find r from the centre of the planet for both points, then substitute

Part 5 of 6Field strength and potential compared

QuantityField strength gPotential Vg
TypeVectorScalar
EquationGM/r2−GM/r
SignTowards the massAlways negative
Falls with r as1/r2−1/r

Part 6 of 6More from the specification

1Gravitational Potential Gradient
  • Defined as the rate of change of gravitational potential with distance,
    • (Delta V / Delta r)
  • where the gravitational field strength equals the negative gradient: g = -dV/dr.
2Orbital Speed of a Satellite
  • For a satellite orbiting Earth at radial distance r, gravitational force provides the centripetal acceleration,
  • resulting in orbital speed v = sqrt(GM / r).
3Escape Speed from a Surface
  • The minimum speed required for an unpowered object at Earth's surface to overcome gravitational attraction completely and reach infinity, given by vesc = sqrt(2GM / R).
4Orbital Energy Conservation
  • A satellite in stable circular orbit possesses negative gravitational potential energy and positive kinetic energy, yielding a bound total mechanical energy of Etotal = -GMm / (2r).
5Kinetic energy
  • K = ½mv2, energy of motion.
6Escape speed
  • Minimum launch speed to reach infinity, √(2GM/R).
7Centripetal force
  • Net force toward the centre, mv2/r.
8Escape velocity
  • Minimum speed to escape a planet's gravity without further propulsion; about 11.2 km/s for Earth.
9Satellite
  • A body that orbits a larger one; the Moon is the Earth's natural satellite.
10Uniform
  • Same density throughout, so the weight acts at the centre.
11Period
  • Time for one revolution.
12Newton
  • Unit of force, kg m s-2.
13Density
  • Mass per unit volume.
14Potential energy
  • Energy stored by the configuration of a system.
15Gravitational force
  • Attractive force between masses.
16Orbital period
  • Time for one orbit.

Key terms

TermMeaning
Potential gradientThe rate of change of gravitational potential with displacement in the direction of the field.
g = −ΔV/ΔrThe field strength equals the negative potential gradient.
TangentA line touching a curve at a point, used to find the gradient there.
Equipotential lineA line joining points of equal gravitational potential.

Avoid these mistakes

1
Forgetting the minus sign in g = −ΔV/Δr.
2
Reading g as the area under the V–r graph — it is the gradient.
3
Using Δr in km without converting to metres.
4
Plotting V positive — gravitational potential is negative.
5
Using the height above the surface instead of the distance from the centre.

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