Question

Mithra is an unknown planet that has two moons, A and B, in circular orbits around it. The table summarizes the

hypothetical data about these moons. What is the mass of Mithra? (G= 6.67 x 10-11N m²/kg2)

Answer 1

To find Mithra's mass, Kepler's third law is applied using Moon A's orbital period and radius. The calculation yields a mass of approximately
`\(3 * 10^(23) \, \text{kg}\)` for Mithra, leading to option (a) as the correct answer.

To determine the mass of Mithra, we can use Kepler's third law, which relates the orbital period (T) and orbital radius (r) of a celestial body to the mass (M) of the central object. The formula is given by:

`\[ T^2 = (4\pi^2 r^3)/(GM) \]`

Let's use the data for Moon A to find the radius of its orbit and then apply it to find the mass of Mithra. First, rearrange the formula to solve for M:

`\[ M = (4\pi^2 r^3)/(GT^2) \]`

For Moon A:

`\[ M_{\text{Mithra}} = \frac{4\pi^2 (2.0 * 10^8 \, \text{m})^3}{G(4.0 * 10^6 \, \text{s})^2} \]`

Now, substitute the values:

`\[ M_{\text{Mithra}} = \frac{4\pi^2 (8.0 * 10^(24) \, \text{m}^3)}{6.67 * 10^(-11) \, \text{N m}^2/\text{kg}^2 * (16 * 10^(12) \, \text{s}^2)} \]`

Calculate the result:

`\[ M_{\text{Mithra}} \approx 3.0 * 10^(23) \, \text{kg} \]`

Therefore, the correct answer is:

a.
`\(3 * 10^(23) \, \text{kg}\).`