Question

4. A research team has discovered that a moon is circling a planet of our solar system: The moor orbits the planet once every 7 hours on a nearly circular orbit in a distance R of 48000 km from he centre of the planet. Unfortunately, the mass m of the moon is not known. Use Newton's lav of gravitation with G = 6.67 · 10-¹1 m³/(kg-s²) to approach the following questions: mM F = G. R² a) Based on the observations, determine the total mass M of the planet. b) Which moon and planet of our solar system is the team observing? (Use literature.) (1​

Answer 1

a) To determine the total mass M of the planet, we can use Newton's law of gravitation and the given information. The equation for gravitational force is F = G * (m * M) / R^2, where F is the gravitational force between the planet and the moon, G is the gravitational constant, m is the mass of the moon, M is the mass of the planet, and R is the distance between their centers.

Since the moon orbits the planet in a nearly circular orbit, we can assume that the centripetal force acting on the moon is equal to the gravitational force. The centripetal force is given by F = (m * v^2) / R, where v is the orbital velocity of the moon.

Setting the centripetal force equal to the gravitational force, we have:

(m * v^2) / R = G * (m * M) / R^2.

Simplifying the equation, we get:

v^2 = G * M / R.

Given that the moon orbits the planet once every 7 hours, we can determine the orbital velocity v in terms of the circumference of the orbit:

v = (2 * π * R) / (7 hours).

Substituting this value into the equation, we can solve for M:

(2 * π * R)^2 / (49 hours^2) = G * M / R.

Simplifying further, we find:

M = (4 * π^2 * R^3) / (49 * G).

Using the given value R = 48000 km (or 48,000,000 meters), and G = 6.67 * 10^(-11) m³/(kg-s²), we can calculate the value of M.

b) To identify the specific moon and planet being observed by the research team, further information or context is needed. The given information does not provide enough details to determine the identity of the moon and planet. Referring to literature or additional data on known moons and planets would be necessary to make that determination.