Final answer:
The centripetal acceleration of the Moon in its orbit around Earth is calculated by using the gravitational force formula and knowing the gravitational constant and the distance between Earth and the Moon. The orbital velocity is not directly required as we use the relationship between gravitational force and centripetal force to find the acceleration.
Step-by-step explanation:
Calculating the Moon's Centripetal Acceleration
To compute the centripetal acceleration of the Moon in its orbit around the Earth, we assume a circular orbit and use the formula for centripetal acceleration (ac) which is ac = v²/r, where v is the orbital velocity of the Moon and r is the radius of the orbit. We need to find the orbital velocity of the Moon, which can be calculated using the formula for gravitational force, where F = G(Mm/r²) and the fact that this force provides the necessary centripetal force (F = m * ac). By combining the two equations, we get ac = G * M/r², where G is the gravitational constant (6.67 × 10⁻¹¹ N. m²/kg²), M is the mass of the Earth, and r is the distance between the centers of the Earth and the Moon (3.84 × 10⁸ meters).
The next step involves substituting the known values into the formula to find the acceleration due to Earth's gravity at the Moon's orbit, which is also the centripetal acceleration required to keep the Moon in its orbit. When comparing this to the acceleration due to Earth's gravity at Earth's surface, it's clear that the values differ due to the variation in distance (the radius of the orbit)