Whenever two Apollo astronauts were on the surface of the...

Whenever two Apollo astronauts were on the surface of the Moon, a third astronaut orbited the Moon. Assume the orbit to be circular and $100 \mathrm{~km}$ above the surface of the Moon. If the mass of the Moon is $7.40 \times 10^{22} \mathrm{~kg}$ and its radius is $1.70 \times 10^{6} \mathrm{~m}$, determine (a) the orbiting astronaut's acceleration, (b) his orbital speed, and (c) the period of the orbit.

Key Concepts

Orbital Period Calculation

The orbital period is the time required for one complete revolution around the central body. It can be calculated using the relation T = 2?r/v or, alternatively, T = 2??(r³/GM). This concept connects the orbital radius and speed, providing insights into how long an orbit takes based on the central mass and the distance of the orbit.

Orbital Speed Determination

The speed of an object in a circular orbit can be found using the relation v = ?(GM/r), where G is the gravitational constant, M is the mass of the central body, and r is the distance from the center of the body to the object. This formula derives from equating the gravitational force to the centripetal force required for circular motion.

Newton's Law of Universal Gravitation

This principle states that any two masses attract each other with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers. It provides the basis for calculating the gravitational pull experienced by an object in orbit, which is essential for determining orbital parameters.

Circular Orbital Motion

In a circular orbit, an object moves at a constant distance from the central body and experiences a centripetal acceleration that keeps it in the circular path. The gravitational force supplies this necessary centripetal force, linking the mass of the central body, the orbital radius, and the velocity of the orbiting object.

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