Calculate the period of a satellite orbiting the Moon,
95 km above the Moon’s surface. Ignore effects of the
Earth. The radius of the Moon is 1740 km.
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To calculate the period of a satellite orbiting the Moon, we can use Kepler's third law, which states that the square of the period of an object orbiting a celestial body is proportional to the cube of the semi-major axis of its orbit.
The semi-major axis of the orbit can be calculated as the sum of the radius of the Moon and the altitude of the satellite above the Moon's surface. In this case, the semi-major axis would be 1740 km + 95 km = 1835 km.
Using the formula for the period of an object in orbit, we can express the relationship as:
T^2 = k * a^3,
where T is the period of the satellite, a is the semi-major axis, and k is a constant of proportionality.
In this case, we can rearrange the formula to solve for the period T:
T = sqrt(k * a^3).
The constant of proportionality (k) can be determined by using the period of the Moon's orbit around the Earth, which is approximately 27.3 days (or 2360591 seconds).
We can plug in the values into the formula and calculate the period:
T = sqrt((2360591 seconds)^2 * (1835 km)^3).
To perform the calculation, we need to convert the units so that they are consistent. We can use the fact that the Moon's average orbital velocity is about 1.022 km/s to convert the period into seconds.
1 km/s ≈ 1000 m/s. So, 1.022 km/s ≈ 1022 m/s.
Thus, the period of the satellite orbiting the Moon, 95 km above its surface, would be:
T = sqrt((2360591 seconds)^2 * (1835 km)^3) / (1022 m/s).
Evaluating the expression will give you the final answer.
To calculate the period of a satellite orbiting the Moon, we can use Kepler's third law of planetary motion. The formula for the period is:
T = 2π√(r^3 / GM)
Where:
T = period of the satellite
π = pi (approximately 3.14159)
r = distance from the center of the Moon to the satellite (in this case, the radius of the Moon plus the altitude of the satellite)
G = gravitational constant
M = mass of the Moon
Given that the radius of the Moon is 1740 km and the satellite is 95 km above the surface, the distance from the center of the Moon to the satellite would be:
r = 1740 km + 95 km = 1835 km
Now we need to find the mass of the Moon and the gravitational constant. The mass of the Moon is approximately 7.348 x 10^22 kg, and the gravitational constant is approximately 6.67430 x 10^-11 m^3/(kg·s^2).
Let's calculate the period:
T = 2π√[(1835 km)^3 / (6.67430 x 10^-11 m^3/(kg·s^2) * 7.348 x 10^22 kg)]
Before we proceed, let's convert the distance and the mass to SI units:
r = 1835 km = 1835000 m
M = 7.348 x 10^22 kg
Now we can substitute the values and calculate the period.