JEE Main 2021 · 27 July · evening · Single correct · +4 / −1

The planet Mars has two moons, if one of them has a period 7 hours, 30 minutes and an orbital radius of 9.0 × 10³ km. Find the mass of Mars.
{ Given (4π ²)/G = 6 × 10¹¹N⁻¹m⁻²kg² }

  1. (a) 5.96 × 10¹⁹ kg
  2. (b) 3.25 × 10²¹ kg
  3. (c) 7.02 × 10²⁵ kg
  4. (d) 6.00 × 10²³ kg
Answer and solution

Answer (d) 6.00 × 10²³ kg

Solution

Physics sir · 4 steps

The moon's orbit weighs the planet: put r and T into Kepler's third law, in metres and seconds.

  1. Gravity gives centripetal force. For a moon of mass m in a circular orbit, GMm/r² = m(4π² r)/T². The moon's mass cancels, so its orbit tells you only about Mars.

    M = 4π²/G·r³/T²

  2. Convert to SI. r = 9.0 × 10³ km = 9.0 × 10⁶ m, so r³ = 729 × 10¹⁸ = 7.29 × 10²⁰ m³. T = 7.5 h = 27000 s, so T² = 7.29 × 10⁸ s².
  3. Put in numbers. r³/T² = (7.29 × 10²⁰)/(7.29 × 10⁸) = 10¹². Then M = 6 × 10¹¹ × 10¹² = 6 × 10²³ kg.
  4. So. The mass of Mars is 6.00 × 10²³ kg, which is option (d).

The idea to keep. Any satellite's r and T give the mass of the body it orbits: M = (4π² r³)/GT². The satellite's own mass never matters. Always convert to metres and seconds first.

Answer from the NTA final answer key. Solution by PrepWiser.

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Asked about 160 times in JEE Main since 2020. This question is from 2021.