Question

An astronaut stands at a position on the Moon such that Earth is directly over head and releases a Moon rock that was in her hand. (a) Which way will it fall? (b) What is the gravitational force exerted by the Moon on a 1.0 -kg rock resting on the Moon's surface? (c) What is the gravitational force exerted by the Earth on the same 1.0 -kg rock resting on the surface of the Moon? (d) What is the net gravitational force on the rock?

Short answer

Answer: The rock will fall towards the lunar surface. The gravitational force exerted by the Moon on the 1.0-kg rock is approximately 1.628 N, the gravitational force exerted by the Earth is approximately 0.0027 N, and the net gravitational force on the rock is approximately 1.625 N.

Step-by-step solution

a) Direction of the fall of the rock

Since the astronaut releases the rock on the Moon, it will fall towards the lunar surface due to the Moon's gravitational force.

b) Gravitational force exerted by the Moon on the rock

To calculate the gravitational force exerted by the Moon, we need to use Newton's Law of Universal Gravitation: F = (G * m1 * m2) / r^2 Where F is the gravitational force, G is the gravitational constant (6.674x10^{-11} N m^2/kg^2), m1 and m2 are the masses of the two objects, and r is the distance between their centers. In this case, m1 = mass of the Moon (7.342x10^22 kg), m2 = mass of the rock (1 kg), and r = radius of the Moon (1.737x10^6 m). F = (6.674x10^{-11} N m^2/kg^2 * 7.342x10^22 kg * 1 kg) / (1.737x10^6 m)^2 F ≈ 1.628 N (approximately) The gravitational force exerted by the Moon on the 1.0-kg rock is approximately 1.628 N.

c) Gravitational force exerted by the Earth on the rock

To find the gravitational force exerted by the Earth, we will use the same formula: m1 = mass of the Earth (5.972x10^24 kg), m2 = mass of the rock (1 kg), and r = distance between the Earth and the Moon (3.844x10^8 m). F = (6.674x10^{-11} N m^2/kg^2 * 5.972x10^24 kg * 1 kg) / (3.844x10^8 m)^2 F ≈ 0.0027 N (approximately) The gravitational force exerted by the Earth on the 1.0-kg rock is approximately 0.0027 N.

d) Net gravitational force on the rock

Since the gravitational forces exerted by the Moon and Earth are in opposite directions, we can calculate the net force by subtracting the smaller force from the larger force: Net force = Force exerted by the Moon - Force exerted by the Earth Net force = 1.628 N - 0.0027 N Net force ≈ 1.625 N (approximately) The net gravitational force on the rock is approximately 1.625 N.