The Lunar Reconnaissance Orbiter (LRO), with mass
m = 1850 kg, maps the surface of the Moon from an orbital altitude of 50.0 km.
(a) What are the magnitude and direction of the force the LRO experiences due to the Moon's gravity?
(b) What are the magnitude and direction of the force exerted by the LRO on the Moon?
To answer these questions, we need to use Newton's law of universal gravitation:
F = G * (m1 * m2) / r^2
where:
- F is the magnitude of the gravitational force between two objects,
- G is the universal gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2),
- m1 and m2 are the masses of the two objects involved, and
- r is the distance between the centers of the two objects.
(a) The magnitude of the force the LRO experiences due to the Moon's gravity:
- mass of the LRO (m1) = 1850 kg
- mass of the Moon (m2) = 7.35 × 10^22 kg
- distance between the centers of the LRO and the Moon (r) = radius of the Moon + orbital altitude = (1,737,400 m + 50,000 m)
Let's calculate this force:
F = G * (m1 * m2) / r^2
F = (6.67430 × 10^-11 m^3 kg^-1 s^-2) * (1850 kg * 7.35 × 10^22 kg) / ((1,737,400 m + 50,000 m)^2)
Performing the calculation will give you the magnitude of the force experienced by the LRO due to the Moon's gravity.
To determine the direction, we consider that the force always acts towards the center of mass of an object. In this case, it will be directed towards the center of the Moon.
(b) The magnitude of the force exerted by the LRO on the Moon:
According to Newton's third law, the force exerted by the LRO on the Moon will be equal in magnitude but opposite in direction to the force experienced by the LRO due to the Moon's gravity.
Therefore, the magnitude of the force exerted by the LRO on the Moon will be the same as the magnitude of the force experienced by the LRO due to the Moon's gravity, and the direction will be in the opposite direction.