The planet Sigma was sighted out in space and was found to have a mass of 8 time

Question

The planet Sigma was sighted out in space and was found to have a mass of 8 times 10^30 kg. Like all of the planets in the Baryon System, it was found to have 3 moons. The first moon, Up, has a mass of 2 times 10^25 kg and is orbiting 4 times 10^10 km away from its planet. Its second moon Down was measured orbiting twice as far as Up, but having a mass twice as large. Finally, the third moon Strange has a mass 4 times that of Up, but orbits 3 times as far away. One evening, while observing this system, a physicist notices that they are curiously aligned perfectly, such that Down and Strange are on its right side, while Up is on its left. The physicist decides to calculate the total net gravitation force on the planet Sigma due to its three moons. Reproduce that physicists calculation (under the strong assumption that it was done correctly...).

Explanation / Answer

The total net gravitational force on the planet Sigma due to its three moons which will be given as :

using a formula, we have

Fnet = G mS mM / r2

where, G = gravitational constant = 6.67 x 10-11 Nm2/kg2

mS = mass of planet Sigma = 8 x 1030 kg

mM = mass of moon = 2 x 1025 kg

r = distance b/w the moon & planet = 4 x 1010 x 103 m

then, we get

Fnet = G mS (mM,1 / r2 + 2mM,2 / 2r2 + 4mM,3 / 3r2)

Fnet = (6.67 x 10-11 Nm2/kg2) (8 x 1030 kg) [(2 x 1025 kg) / (4 x 1013 m)2 + (2 x 2 x 1025 kg) / (2 x 4 x 1013 m)2 + (4 x 2 x 1025 kg) / (3 x 4 x 1013 m)2]

Fnet = (53.36 x 1019 Nm2/kg) (2 x 1025 kg) / (4 x 1013 m)2 [1 + 1 + 4/3]

Fnet = (6.67 x 1018 N) (10/3)

Fnet = 2.23 x 1019 N

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