Universal gravitation 5- determine which satellite has smaller period

AI Thread Summary
The discussion focuses on determining which satellite, orbiting Neptune or Earth, has a smaller orbital period based on their respective masses and orbital radii. Using Kepler's third law and gravitational equations, calculations show that Neptune's satellite, with a mass approximately 17.2 times that of Earth, has a shorter orbital period of about 1265.82 seconds compared to Earth's satellite, which has a period of 5249.72 seconds. The calculations confirm that the satellite orbiting Neptune has the smaller period due to the greater mass of Neptune affecting the gravitational force. Participants are encouraged to verify the calculations and understand the relationship between a planet's mass and its satellite's orbital period. The conclusion is that Neptune's satellite indeed has a smaller period.
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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: (Ta/Tb)2=(Ra/Rb)3

motion of planets must conform to circular motion equation: Fc=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∏2Ks)*m/R2=Gmsm/R2

Gravitational force of attraction between the Earth and its orbiting satelittes: F=(4∏2Ke)m/R2=Gmem/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=Fg=mog, where g=9.8 N/kg
Fg=Gmome/Re2

g=Gme/(Re)2

determine the mass of the Earth: me=g(Re)2/G

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

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

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

Determine mass of planet when given orbital period and mean orbital radius: Mp=4∏2Rp3/GTp2



The Attempt at a Solution



mN=17.2x me
=17.2x(5.98x1024kg)=1.02856x1026kg

RN=Re=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! :)
 
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You can just try to find the relation between the mass of planet and the period of its satellite without doing the numerical calculation.
 
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