Universal gravitation 5- determine which satellite has smaller period

In summary: The mass of the planet is proportional to the period of its satellite, so the satellite with the smaller period will have a larger mass. Therefore, the satellite orbiting Neptune will have a smaller period and a larger mass compared to the satellite orbiting Earth. In summary, the satellite orbiting Neptune has a smaller period and a larger mass compared to the satellite orbiting Earth, based on the given information and relevant equations.
  • #1
dani123
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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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  • #2
You can just try to find the relation between the mass of planet and the period of its satellite without doing the numerical calculation.
 

1. How is universal gravitation related to satellites?

Universal gravitation is the force of attraction between any two objects with mass. This force allows satellites to stay in orbit around larger celestial bodies, such as planets.

2. How does the mass of a satellite affect its period of orbit?

The mass of a satellite does not affect its period of orbit. The period of orbit, or the time it takes for a satellite to complete one full revolution around a celestial body, is only determined by the distance between the satellite and the body it is orbiting.

3. Can you determine which satellite has a smaller period of orbit?

Yes, the period of orbit can be determined by using Kepler's Third Law, which states that the square of the period of orbit is proportional to the cube of the distance between the satellite and the celestial body it is orbiting. Therefore, the satellite with a smaller distance will have a smaller period of orbit.

4. How is the period of orbit affected by the strength of universal gravitation?

The strength of universal gravitation does not affect the period of orbit. The period of orbit is only determined by the distance between the satellite and the celestial body it is orbiting, as stated in Kepler's Third Law.

5. Can a satellite have a period of orbit longer than that of its celestial body?

Yes, it is possible for a satellite to have a longer period of orbit than its celestial body. This can occur if the satellite is orbiting at a greater distance from the body than the distance between the body and its own satellite. For example, the Moon has a longer period of orbit than Earth, as it orbits at a greater distance from the Sun than Earth does.

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