The orbit of a Moon about its planet is approximately circular, with a mean radius of
2.24 × 10
8 m. It takes 35.5 days for the Moon
to complete one revolution about the planet.
Find the mean orbital speed of the Moon.
Answer in units of m/s
Find the Moon’s centripetal acceleration.
Answer in units of m/s
2
The mean orbital speed of the Moon is:
(2.24 x 10^8 m) / (35.5 days x 24 hours/day x 60 minutes/hour x 60 seconds/minute) = 1.022 x 10^3 m/s
The Moon's centripetal acceleration is:
(1.022 x 10^3 m/s)^2 / (2.24 x 10^8 m) = 4.5 x 10^-5 m/s^2
To find the mean orbital speed of the Moon, we can use the formula:
v = 2πr / T
where v is the mean orbital speed, r is the mean radius of the orbit, and T is the time taken to complete one revolution.
Substituting the given values:
r = 2.24 × 10^8 m
T = 35.5 days = 35.5 * 24 * 60 * 60 seconds
Let's calculate the mean orbital speed:
v = (2 * π * 2.24 × 10^8) / (35.5 * 24 * 60 * 60)
v ≈ 1020.31 m/s
Therefore, the mean orbital speed of the Moon is approximately 1020.31 m/s.
To calculate the Moon's centripetal acceleration, we can use the formula:
a = v^2 / r
Substituting the values:
v = 1020.31 m/s
r = 2.24 × 10^8 m
Let's calculate the centripetal acceleration:
a = (1020.31)^2 / (2.24 × 10^8)
a ≈ 4668.91 m/s^2
Therefore, the Moon's centripetal acceleration is approximately 4668.91 m/s^2.
To find the mean orbital speed of the Moon, we can use the formula:
Mean Orbital Speed = (2 * π * R) / T
where:
- R is the mean radius of the orbit
- T is the period of the orbit
Given that the mean radius of the orbit is 2.24 × 10^8 m and the period is 35.5 days, we need to convert the period from days to seconds. There are 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute:
Conversion: 35.5 days x 24 hours/day x 60 minutes/hour x 60 seconds/minute = T seconds
Now we can calculate the mean orbital speed:
Mean Orbital Speed = (2 * π * (2.24 × 10^8)) / T
Plug in the values and calculate the mean orbital speed.
To find the Moon's centripetal acceleration, we can use the formula:
Centripetal Acceleration = (v^2) / R
where:
- v is the orbital speed
- R is the mean radius of the orbit
Given that we already calculated the mean orbital speed as v, we can simply plug in the values and calculate the centripetal acceleration.
Make sure to perform all calculations using the correct order of operations (e.g., parentheses, exponents) and use the appropriate units throughout the calculations.