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A2 · Practice questions · Orbits and field strength

Gravity alone is the centripetal force.

Six original Cambridge-style questions on circular orbits, orbital speed and period, the geostationary condition and the field strength g = GM / r² of a point mass.

Original questions All questions on this page are original work, written in the Cambridge AS & A Level style. They are not from past papers. They test the same concepts and skills the syllabus rewards.
Keep these straight

Equate gravity to the centripetal force.

01
Analysis
[2 marks]

A satellite is in a circular orbit around the Earth. Explain why it is incorrect to say that the satellite stays in orbit because "gravity is balanced by the centripetal force".

  • There is only one force on the satellite, the Earth's gravitational pull; it is not balanced by anything, so the satellite is not in equilibrium but accelerating toward the Earth's centre. ✓
  • The centripetal force is not a separate force: it is the name for the resultant force needed for circular motion, and here gravity itself provides it, GMm / r² = mv² / r. ✓
02
Calculation
[3 marks]

A space station orbits the Earth (mass 6.0 × 10²⁴ kg) in a circular orbit of radius 6.8 × 10⁶ m. Calculate its orbital speed and the time it takes to complete one orbit, in minutes.

  • v = √(GM / r) = √((6.67 × 10⁻¹¹ × 6.0 × 10²⁴) / 6.8 × 10⁶) = √(5.89 × 10⁷). ✓
  • v = 7.7 × 10³ m s⁻¹. ✓
  • T = 2πr / v = (2π × 6.8 × 10⁶) / 7.67 × 10³ = 5.57 × 10³ s = 93 minutes. ✓
03
Calculation
[3 marks]

Io, a moon of Jupiter, moves in a circular orbit of radius 4.22 × 10⁸ m with a period of 1.77 days. Use these data to calculate the mass of Jupiter.

  • Gravity provides the centripetal force: GMm / r² = mrω² = mr(2π / T)², so M = 4π²r³ / (GT²). ✓
  • T = 1.77 × 24 × 3600 = 1.53 × 10⁵ s. ✓
  • M = (4π² × (4.22 × 10⁸)³) / (6.67 × 10⁻¹¹ × (1.53 × 10⁵)²) = 1.9 × 10²⁷ kg. ✓
04
Analysis
[3 marks]

A satellite is placed in a circular orbit over the Earth's poles with a period of exactly 24 hours. Explain why this satellite is not geostationary, and explain why every geostationary satellite must orbit at the same height.

  • A geostationary satellite must stay above one fixed point on the Earth's surface, which requires it to orbit in the equatorial plane in the same direction as the Earth's rotation; a polar orbit crosses the equator and sweeps over different ground as the Earth turns beneath it. ✓
  • The period is fixed by the radius alone, T = 2π√(r³ / GM), since the satellite mass cancels from GMm / r² = mrω². ✓
  • A period of 24 h therefore corresponds to one radius only (about 4.2 × 10⁷ m), so all geostationary satellites share that height. ✓
05
Calculation
[3 marks]

A satellite orbits the Earth (mass 6.0 × 10²⁴ kg) at a distance of 1.2 × 10⁷ m from the centre. Calculate the gravitational field strength at this distance, state the centripetal acceleration of the satellite, and hence calculate its orbital speed.

  • g = GM / r² = (6.67 × 10⁻¹¹ × 6.0 × 10²⁴) / (1.2 × 10⁷)² = 2.78 N kg⁻¹. ✓
  • The only force is gravity, so the centripetal acceleration equals the field strength: a = 2.78 m s⁻² toward the Earth's centre. ✓
  • a = v² / r gives v = √(ar) = √(2.78 × 1.2 × 10⁷) = 5.8 × 10³ m s⁻¹. ✓
06
Calculation
[4 marks]

Mars has mass 6.42 × 10²³ kg and rotates once on its axis every 24.6 hours. A communications satellite is to be placed in orbit so that it remains above the same point on the Martian equator.

(a) State the condition on the period of the satellite's orbit. (b) Show that the radius of the orbit is about 2.0 × 10⁷ m. (c) Calculate the orbital speed of the satellite.

  • (a) The orbital period must equal the rotation period of Mars, 24.6 h = 8.86 × 10⁴ s, with the satellite moving in the equatorial plane in the same sense as the planet's rotation. ✓
  • (b) GMm / r² = mr(2π / T)², so r³ = GMT² / 4π² = (6.67 × 10⁻¹¹ × 6.42 × 10²³ × (8.86 × 10⁴)²) / 4π² = 8.51 × 10²¹ m³. ✓
  • r = (8.51 × 10²¹)1/3 = 2.04 × 10⁷ m, about 2.0 × 10⁷ m. ✓
  • (c) v = 2πr / T = (2π × 2.04 × 10⁷) / 8.86 × 10⁴ = 1.4 × 10³ m s⁻¹. ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.