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Circular motion and gravitational fields (A Level)

13. Gravitational Fields

5 min read 10 parts Lesson 14 of 14 Unit quiz

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13. Gravitational Fields — Cambridge AS & A Level Physics 9702 — CAMBRIDGE AS & A LEVEL PHYSICS 9702 — COMPLETE NOTES notes: Apply Newton's law of gravitation, field strength, orbital motion, geostationary orbits, gravitational potential and potential energy to planets and satell…

  • Gravitational field strength g
  • Near the Earth's surface
  • Field lines
  • Planets and stars
  • Separation
  • Inverse square law

Section 1 of 10Gravitational field and field strength

  • Gravitational field = a region of space where a mass experiences a force.
    • The force is always attractive and points towards the centre of the mass producing it.
  • Gravitational field strength g = the force per unit mass acting on a small mass at that point, g = F/m, in N kg−1.
  • Weight
    • a planet with a larger g gives objects a larger weight, since weight = mg.
  • Notation
    • g (field strength) is not G.
      • (Newton's gravitational constant)

Section 2 of 10Field lines

Field typeField linesField strength
Radial (around a point mass or sphere)Directed inwards towards the centre; they spread out with distanceg decreases with distance from the centre
Uniform (near a planet's surface)Equally spaced parallel lines towards the surfaceg is the same everywhere
  • Near the Earth's surface
    • the field is almost uniform and can be treated as uniform.
  • Arrows
    • field lines carry arrows because gravitational forces are attractive only.

Section 3 of 10Point-mass approximation

  • Rule
    • for a point outside a uniform sphere, all the mass can be treated as concentrated at its centre.
    • An object can be a point mass when its mass distribution is even and the distance considered is larger than its size.
  • Field lines
    • those around a uniform sphere are identical to those around a point mass.
  • Planets and stars
    • they qualify because they are nearly uniform spheres and their separations are much greater than their radii.

Section 4 of 10Newton's law of gravitation

  • Newton's law = 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, with G = 6.67 × 10−11 N m2 kg−2.
      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
    • r is measured between the centres of the masses.
    • For a satellite above a planet r = planet radius + height above the surface.
  • Inverse square law
    • doubling the separation reduces the force to one quarter.

Section 5 of 10Field strength due to a point mass

  • Derivation
    • the force on a small mass m at distance r from mass M is F = GMm/r2, and the definition gives F = mg.
    • Equating and cancelling m gives g = GM/r2.
  • Vector
    • g points towards the centre of the body.
  • Inverse square
    • g ∝ 1/r2, so it falls quickly as r increases.
  • Two equations
    • g = F/m is the definition at a point, and g = GM/r2 is the field of a point mass.

Section 6 of 10The value of g near the Earth

  • Near the surface
    • g is about 9.81 N kg−1 and is treated as constant, since heights h in the atmosphere are tiny compared with the Earth's radius R.
    • At h = 150 km, R + h = 6.55 × 106 m, only slightly more than R = 6.4 × 106 m.
  • General form
    • g = GM/(R + h)2, which reduces to GM/R2 at the surface.
  • Density form
    • for a sphere M = ρ × (4/3)πr3, so g = 4πGρr/3, useful for comparing planets.

Section 7 of 10Circular orbits

  • Centripetal force
    • the gravitational force on an orbiting body provides it: GMm/r2 = mv2/r.
  • Orbital speed
    • the satellite mass cancels, so v = √(GM/r), and all satellites at one radius have the same speed.
  • Direction
    • the velocity is along the tangent and the force towards the centre, so the velocity changes continually.
  • Kepler's third law
    • substituting v = ωr gives ω2r3 = GM, and with ω = 2π/T this gives T2 ∝ r3.
  • Binary stars
    • each star's gravitational force provides its centripetal force, and both stars have the same ω.

Section 8 of 10Geostationary orbits

  • Properties
    • a geostationary satellite stays above the equator at a fixed point on the Earth.
    • It moves from west to east, the same way as the Earth turns.
    • Its period is 24 hours, equal to the Earth's rotation period.
  • Orbit radius
    • setting T = 24 h in T2 = 4π2r3/GM fixes the radius.
  • Uses
    • telecommunications and television broadcasting.
    • A ground station sends a signal up, the satellite amplifies it and sends it to chosen areas.
    • Receiver dishes point at a fixed position in the sky and need not move.

Section 9 of 10Gravitational potential

  • Gravitational potential φ = the work done per unit mass in bringing a small test mass from infinity to the point.
    • It is zero at infinity and negative everywhere else, because gravity is attractive.
  • Equation
    • φ = −GM/r, in J kg−1, which depends on the mass and distance from its centre.
    • It becomes less negative as r increases and decreases as a satellite moves closer.
  • Change in potential
    • Δφ = GM(1/r1 − 1/r2) between distances r1 and r2.

Section 10 of 10Gravitational potential energy

  • Work done
    • moving a mass m through a potential difference Δφ needs work W = mΔφ.
  • Gravitational potential energy = the work done to bring the masses from infinite separation to distance r, Ep = −Gm1m2/r.
  • Change
    • ΔEp = Gm1m2(1/r1 − 1/r2), the work done against gravity from r1 to r2.
  • Limit of mgh
    • Ep = mgh holds only in a uniform field near a surface; elsewhere use −GMm/r.

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