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Theme D — Fields

D.1 Gravitational Fields — IB Physics HL Revision Notes & Topical Questions

5 min read 12 parts Lesson 2 of 39 1 quiz

📚 IB Physics HL Hub / Topical Revision
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pH = -log[H⁺] | pOH = -log[OH⁻] | pH + pOH = 14 | [H⁺][OH⁻] = 10⁻¹⁴
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Lesson summary

D.1 Gravitational Fields — IB PHYSICS HL notes: Apply Newton's law of gravitation and Kepler's laws, and use gravitational field strength, potential, potential energy, escape speed and orbital energy.Statement — the force between two point masses is proportional to the product of…

  • What the symbols mean
  • Separation
  • Inverse square
  • Definition
  • Point or spherical mass
  • Dependence

Part 1 of 10Newton's law of gravitation

Statement
  • Statement the force between two point masses is proportional to the product of the masses and inversely proportional to the square of their separation
  • F = Gm1m2/r2
    What the symbols mean: F = gravitational force (N) · G = 6.67 × 10−11 N m2 kg−2 · m1, m2 or M, m = masses (kg) · r = distance between their centres (m)
Separation
  • Separation r is measured between centres, so for a satellite add the planet radius to the altitude
Point masses
  • Point masses planets and stars are treated as points, and a uniform sphere acts as if all its mass were at its centre
Inverse square
  • Inverse square doubling the distance reduces the force to a quarter
Constant
  • G = 6.67 × 10−11 N m2 kg−2

Part 2 of 10Gravitational field strength

Field
  • Field a region where a test mass experiences a force due to the gravitational attraction of another mass
Definition
  • Definition g = F/m, the force per unit mass, in N kg−1
Point or spherical mass
  • Point or spherical mass g = GM/r2 from equating mg with GMm/r2
Dependence
  • Dependence g depends on the mass producing the field and r, not on the test mass
Vector
  • Vector it points towards the mass, and fields from several masses add as vectors
g and G
  • g and G g is the field strength and G is Newton's gravitational constant

Part 3 of 10Gravitational field lines

Gravitational field lines
Direction
  • Direction field lines point towards the centre of the mass because gravity is attractive only
Point mass or uniform sphere
  • Point mass or uniform sphere radial lines, which make a non-uniform field with g varying with distance
Near the surface
  • Near the surface parallel equally spaced lines make a uniform field
Spacing
  • Spacing closer lines mean a stronger field
Labelling
  • Labelling always put arrows on field lines

Part 4 of 10Kepler's laws and circular orbits

1First law
  • First law each planet moves in an ellipse with the Sun at one focus
2Second law
  • Second law the line from Sun to planet sweeps out equal areas in equal times
  • A planet moves fastest when closest to the Sun
3Third law
  • T2 ∝ r3 for bodies orbiting the same central mass
4Orbital speed
  • GMm/r2 = mv2/r gives v = √(GM/r), independent of the satellite's mass
5Period
  • Period with v = 2πr/T, T2 = 4π2r3/GM
6Log plot
  • Log plot log T against log r is a straight line with a gradient of 3/2

Part 5 of 10Gravitational potential

Definition
  • Definition the work done per unit mass in bringing a test mass from infinity to the point
  • Vg = −GM/r in J kg−1
Zero at infinity
  • Zero at infinity it is negative elsewhere because work must be done to move a mass away from the body
Scalar
  • Scalar contributions add algebraically
Distance
  • Distance potential becomes less negative as a mass moves away and is more negative closer in
M
M only the mass producing the field appears in the equation

Part 6 of 10Potential energy and work

Radial field
  • Ep = mVg = −GMm/r, the work to bring a mass from infinity
Change
  • ΔEp = GMm(1/r1 − 1/r2)
  • ΔVg = GM(1/r1 − 1/r2) for a move from r1 to r2, negative when the mass moves closer
Work
  • W = mΔVg
  • Near the surface this is W = mgΔh
Force–distance graph
  • Force–distance graph the area under it is the change in potential energy
mgh
  • mgh valid only near the surface where the field is uniform

Part 7 of 10Potential gradient and equipotentials

Gradient
  • Gradient g = −ΔVg/Δr, the negative gradient of the potential–distance graph
V–r graph
  • V–r graph a −1/r curve with all values negative
  • Draw a tangent and take its gradient to find g
Equipotentials
  • Equipotentials surfaces of equal potential, perpendicular to field lines and drawn dotted without arrows
Radial field
  • Radial field concentric circles that get further apart away from the planet
Uniform field
  • Uniform field horizontal, parallel and equally spaced lines
Along an equipotential
  • Along an equipotential no work is done

Part 8 of 10Orbital energy

Kinetic energy
  • Ek = GMm/2r
Potential energy
  • Ep = −GMm/r
Total energy
  • E = −GMm/2r, negative for a bound orbit
Smaller orbit
  • Smaller orbit larger force, speed and Ek, lower Ep and shorter period
Larger orbit
  • Larger orbit smaller force, speed and Ek, higher Ep and longer period

Part 9 of 10Escape speed and drag

Escape speed
  • Escape speed the minimum speed to escape a gravitational field with no further energy input
  • ½mv2 = GMm/r gives vesc = √(2GM/r) = √2 × orbital speed
Independence
  • Independence it is the same for all masses and directions
Drag
  • Drag viscous drag from a thin atmosphere removes total energy in low orbits
    • (below about 600 km)
  • The orbital radius decreases and the satellite speeds up

Part 10 of 10More from the specification

1Radial gravitational field potential
  • Radial gravitational field potential evaluating work required to move unit masses from infinity provides a calculus-based means to define gravitational potential.
2Newton's Law of Universal Gravitation
  • The attractive gravitational force acting between two spherical bodies of mass M and m separated by distance r is given by F = G * M * m / r2, directed toward their centres of mass.
3Orbital Motion and Centripetal Force
  • In circular orbital motion of a satellite orbiting Earth at distance r, the gravitational force provides the necessary centripetal force,
    • (G * M * m / r2 = m * v2 / r)
  • leading to orbital speed v = sqrt(G * M / r).
4Escape Velocity from Earth's Surface
  • The escape velocity represents the minimum initial speed an object must have at the surface of Earth to escape gravitational pull without further propulsion, derived by energy conservation as vesc = sqrt(2 * G * M / R).
    • (at Earth radius R)
5Gravitational Field Strength and Radial Variation
  • Gravitational field strength g = GM / r2 decreases with distance r; for a planet of uniform density rho, g increases linearly with radius inside the planet before following an inverse-square decline outside.
    • (g proportional to r)
6Density
  • Mass per unit volume.
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.

Key terms

TermMeaning
Gravitational field strengthThe force per unit mass on a small test mass placed at the point.
Newton's law of gravitationEvery point mass attracts every other with a force proportional to the product of the masses and inversely proportional to the square of their separation.
Field lineA line whose direction at each point shows the direction of the force on a test mass.
Orbital periodThe time for one complete orbit.
Gravitational potentialThe work done per unit mass in bringing a small test mass from infinity to the point.
Equipotential surfaceA surface on which the potential is the same at every point.
Escape speedThe minimum speed needed for a body to escape from a gravitational field to infinity.

Avoid these mistakes

1
Measuring r from the surface instead of the centre.
2
Using the satellite's mass in the orbital speed formula — it cancels.
3
Saying a higher orbit needs a higher speed — orbital speed is lower for a larger radius.
4
Giving a positive total energy for a bound orbit.
5
Confusing g with G.

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