Question

The Moon and Earth rotate about their common center of mass, located about 4700 km from the center of Earth. Calculate the acceleration due to the Moon’s gravity at that point. Calculate the centripetal acceleration of the center of Earth as it rotates about that point once each lunar month (about 27.3 d) and compare it with the acceleration found in part a. Comment on whether or not they are equal and why they should or should not be.

a) Higher, Lower, They are not equal due to the rotation
b) Higher, Higher, They are equal due to the balance of forces
c) Lower, Lower, They are equal due to the balance of forces
d) Lower, Higher

Answer 1

The acceleration due to the Moon's gravity at the point where the Moon and Earth rotate about their common center of mass is approximately 0.0027 m/s^2. The centripetal acceleration of the center of Earth as it rotates about that point is approximately 2.22 × 10^-7 m/s^2. These accelerations are not equal because the force of gravity exerted by the Moon on Earth causes the center of Earth to rotate about their common center of mass, resulting in a higher acceleration.

Step-by-step explanation:

(a) The acceleration due to the Moon's gravity at the point where the Moon and Earth rotate about their common center of mass can be calculated using the formula for gravitational acceleration:

g = G * (m/r^2)

where G is the gravitational constant (6.67430 × 10^-11 Nm^2/kg^2), m is the mass of the Moon (7.35 × 10^22 kg), and r is the distance from the center of Earth to the center of the Moon (4700 km + 1690 km = 6390 km = 6.39 × 10^6 m).

Plugging in the values, we get:

g = (6.67430 × 10^-11 Nm^2/kg^2) * (7.35 × 10^22 kg)/(6.39 × 10^6 m)^2

Solving for g gives:

g ≈ 0.0027 m/s^2

(b) The centripetal acceleration of the center of Earth as it rotates about the common center of mass can be calculated using the formula for centripetal acceleration:

ac = v^2/r

where v is the velocity of the center of Earth and r is the distance from the center of Earth to the common center of mass.

Since the center of Earth rotates once each lunar month (27.3 days), we can calculate the velocity using:

v = 2πr/T

where T is the time period of rotation (27.3 days = 27.3 * 24 * 3600 seconds).

Plugging in the values, we get:

v = 2π(6.39 × 10^6 m)/(27.3 * 24 * 3600 s)

Solving for v gives:

v ≈ 1.19 m/s

Now, plugging in the values of v and r into the formula for centripetal acceleration, we get:

ac = (1.19 m/s)^2/(6.39 × 10^6 m)

Solving for ac gives:

ac ≈ 2.22 × 10^-7 m/s^2

Comparing the acceleration due to the Moon's gravity (0.0027 m/s^2) to the centripetal acceleration of the center of Earth (2.22 × 10^-7 m/s^2), we can see that they are not equal. The acceleration due to the Moon's gravity is higher than the centripetal acceleration of the center of Earth.

This is because the force of gravity exerted by the Moon on Earth causes the center of Earth to rotate about their common center of mass, resulting in a higher acceleration compared to the centripetal acceleration needed to keep the Moon in its orbit.