Final answer:
The period for the moon's motion around the earth is approximately 0.59 days, which is much shorter than the length of a month.
Step-by-step explanation:
To find the period for the moon's motion around the earth, we can use Kepler's third law. According to Kepler's third law, the square of the period of a planet's orbit is directly proportional to the cube of its average distance from the center of the orbit.
We are given that the moon orbits the earth at a distance of 3.85 x 10^8 m. We can use this information to calculate the period as follows:
- Convert the given distance to meters: 3.85 x 10^8 m.
- Calculate the period using Kepler's third law equation:
T^2 = (4π^2/GM) * r^3
where T is the period, G is the gravitational constant (6.67430 × 10^-11 m^3 kg^-1 s^-2), M is the mass of the earth (5.98 x 10^24 kg), and r is the distance between the centers of the earth and the moon. - Substitute the known values into the equation and solve for T:
T^2 = (4π^2/(6.67430 × 10^-11 m^3 kg^-1 s^-2)) * (5.98 x 10^24 kg) * (3.85 x 10^8 m)^3
T^2 ≈ 2.97 x 10^7 s^2
T ≈ √(2.97 x 10^7) s ≈ 5.14 x 10^3 s. - Convert the period from seconds to days:
1 day = 24 hours × 60 minutes × 60 seconds = 86,400 seconds.
T ≈ 5.14 x 10^3 s / 86,400 s/day ≈ 0.59 days.
Hence, the period for the moon's motion around the earth is approximately 0.59 days. This is much shorter than the length of a month, which is about 30 days. Therefore, the moon completes multiple orbits around the earth in one month.