Planet X rotates in the same manner as the earth, around an axis through its nor
ID: 1462919 • Letter: P
Question
Planet X rotates in the same manner as the earth, around an axis through its north and south poles, and is perfectly spherical. An astronaut who weighs 945.0 N on the earth weighs 915.0 N at the north pole of Planet X and only 849.0 N at its equator. The distance from the north pole to the equator is 1.883×104 km , measured along the surface of Planet X.
A. How long is the day on Planet X?
B. If a 4.200×104 kg satellite is placed in a circular orbit 3000 km above the surface of Planet X, what will be its orbital period?
Explanation / Answer
Apply the law of gravitation to the astronaut at the north pole to calculate the mass of planet
radius = 2piR/4 = 1.883 x 10^4 km
R = 1.1988 x 10^7 m
astronaut mass = w/g = 945/9.8 = 96.4 kg
Gm*Mx/R^2 = w
w = 915 N
Mx = mgx * R^2 /Gm = 2.04 x 10^25 kg
Fgrav - Fscale = 4*pi^2 *mR/T^2
915 - 849 = 4*pi^2 *mR/T^2
66 = 4 * pi^2 * 96.4 * 1.1988 x 10^7/T^2
T = 7.303 h
part b )
for the satellite
T = sqrt(4pi^2 * r^3 /GMx)
here r = 1.1988 x 10^7 + 3000 x 10^3 = 14988000
T = 2.745 h