1. a. A satellite is in a circular orbit around the earth. The period of the sat
ID: 1428022 • Letter: 1
Question
1. a. A satellite is in a circular orbit around the earth. The period of the satellite is 23.7 hr. Calculate the radius of the orbit of the satellite. Data: Mass of the earth = 5.98 x 1024 kg.
b. What is the speed of the satellite (in m/s)?
2. An astronaut weighs 662 N on the Earth. What is her weight on planet X, which has a radius Rx = Rearth / 2.20 and a mass Mx = Mearth/7.90?
3. The asteroid Icarus orbits the Sun like the other planets. Its period is about 410 days. What is its mean distance from the Sun? (Note: this problem does not involve much calculator use, IF you set up a ratio, and use time-units of days and distance units of AU. 1 AU is the distance from the Earth to the Sun. CAPA will accept units of AU for distance.)
Explanation / Answer
Period of satellite = 23.7 hours = 85,320 seconds
Mass of Earth, M = 5.98 x 10^24 kg
G = 6.67 x10^ -11 m^3 kg ^ - 1 s^ -2
Use Kepler's third law modified by Newton
P^2 = (2)^2(A^3)/(G(M)) where P is the period in seconds, A is the distance in meters, M is the mass in kg, and G is the gravitational constant
A^3 = (G)(M)(P^2)/((2)^2
A^3 = (6,67x 10^ -11)(5.98x 10^24)(8.532 x 10^4)^2/(39.5)
A = 4.19 x 10^7 meters
A is 41,900 kilometers from the center and since the Earth has a radius of 6,380 kilometers
The satellite orbits 35,500 kilometers above the surface
2.gravitational acceleration g = GM/ R^2
as M' = M/7
and R' = R/2.2
=> g ' = G (M/7)/ (R/2.2)^2
= 0.454 g
So it weight will be = 662 * 0.454 = 300.548 N
3. Msun = 2.0 * 10^30 kg
G = 6.673 * 10^-11 N•m^2/kg^2
T= 410 days = 35424000 s
T^2/ R^3= 4* pi^2 / (G*Msun)
T^2/ R^3= 2.96*10^-19
R = 161,845,698,924 m
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