Universal gravitation 5- determine which satellite has smaller period

[dani123]

Homework Statement

Neptune is about 17.2 times the mass of Earth. Satellite N orbits Neptune with the same orbital radius as satellite E that orbits the Earth. Determine which satellite has the smaller period. Support your answer with appropriate calculations.

Homework Equations

I have made a list of equations that are relevant for this entire module on universal gravitation. So although there are many of them does not mean that they all apply in this circumstance. The ones relevant to this question will be placed in bold.

Kepler's 3rd law: (T_a/T_b)²=(R_a/R_b)³

motion of planets must conform to circular motion equation: F_c=4∏²mR/T²

From Kepler's 3rd law: R³/T²=K or T²=R³/K

Gravitational force of attraction between the sun and its orbiting planets: F=(4∏²K_s)*m/R²=Gm_sm/R²

Gravitational force of attraction between the Earth and its orbiting satelittes: F=(4∏²K_e)m/R²=Gm_em/R²

Newton's Universal Law of Gravitation: F=Gm₁m₂/d²

value of universal gravitation constant is: G=6.67x10^-11N*m²/kg²

weight of object on or near Earth: weight=F_g=m_og, where g=9.8 N/kg
F_g=Gm_om_e/R_e²

g=Gm_e/(R_e)²

determine the mass of the Earth: m_e=g(R_e)²/G

speed of satellite as it orbits the Earth: v=√GM_e/R, where R=R_e+h

period of the Earth-orbiting satellite: T=2∏√R³/GM_e

Field strength in units N/kg: g=F/m

Determine mass of planet when given orbital period and mean orbital radius: M_p=4∏²R_p³/GT_p²

The Attempt at a Solution

m_N=17.2x m_e
=17.2x(5.98x10²⁴kg)=1.02856x10²⁶kg

R_N=R_e=6.53x10⁶m

For Neptune satellite:

T=2∏√(6.53X10⁶)³/(6.67X10^-11)(1.029X10²⁶)
T=1265.82

For Earth satellite:
T=2∏√(6.53x10⁶m)³/(6.67x10^-11)(5.98x10²⁴kg)
T=5249.72

∴Neptune's satellite has the smaller period.
If anyone could please verify my work and let me know if/where I went wrong that would be greatly appreciated! Thanks so much for you help! :)

 

[grzz]

You can just try to find the relation between the mass of planet and the period of its satellite without doing the numerical calculation.