Final answer:
To calculate the centripetal acceleration of Earth's center during a lunar month, use Earth’s orbital velocity around the common center of mass with the Moon. This calculated acceleration will differ from the acceleration due to the Moon’s gravity, as they describe different dynamics within the Earth-Moon system.
Step-by-step explanation:
To calculate the magnitude of the centripetal acceleration of Earth's center as it rotates about the common center of mass with the Moon once each lunar month (about 27.3 days), we need to determine the velocity of Earth around this point and then use the formula ac = v² / r. The common center of mass is 4700 km from the center of Earth. We would need the radius of the orbit and the time period to find the velocity of Earth's center around the common center of mass.
Once we have the centripetal acceleration, we can compare it with the acceleration due to the Moon's gravity at that point, which was calculated in part (a). The two accelerations are expected not to be equal because they describe different physical situations: part (a) refers to the gravitational pull while this part refers to the centripetal force required for Earth's rotation around the common center of mass.
The similarity or difference between these accelerations will provide insight into the dynamics of the Earth-Moon system. Typically, the centripetal acceleration of Earth in orbit due to its revolution around a common center of mass with the Moon will be much less than the acceleration due to the Moon's gravity at Earth's surface since the forces responsible for the acceleration and the context of the two scenarios are different.