Problem
Find an approximate value for the rate at which the moon orbits the earth. Assume that the moon’s orbit is circular.
To find an approximate value for the rate at which the moon orbits the earth, we need to know the time it takes for the moon to complete one orbit around the earth. This is also known as the orbital period of the moon.
The orbital period of the moon can be determined by using Kepler's Third Law, which states that the square of the orbital period is directly proportional to the cube of the average distance between the earth and the moon.
The average distance between the earth and the moon is approximately 384,400 kilometers. We can use this information to find the orbital period.
1. Convert the average distance between the earth and the moon to meters: 384,400 kilometers = 384,400,000 meters.
2. Use the formula for the orbital period: T^2 = (4π^2 * r^3) / (GM), where T is the orbital period, r is the average distance between the earth and the moon, G is the gravitational constant (approximately 6.67 x 10^-11 m^3 / (kg s^2)), and M is the mass of the earth (approximately 5.97 x 10^24 kg).
3. Rearrange the formula to solve for T: T = √((4π^2 * r^3) / (GM)).
4. Substitute the values into the formula: T = √((4 * 3.14^2 * (384,400,000)^3) / ((6.67 x 10^-11) * (5.97 x 10^24))).
5. Simplify the equation and calculate the result.
By following these steps, you can approximate the rate at which the moon orbits the earth based on its orbital period.