Universal gravitation 1-determine height of satellite

In summary, you calculated the height of the second satellite above the Earth's surface as 8998294.4 meters.
  • #1
dani123
136
0

Homework Statement



Two satellites are orbiting around the Earth. One satellite has a period of 1.5 h and is 250 km above the Earth's surface. The second satellite has a period of 7.5 h. Using Kepler's Laws and the fact that the Earth's radius is 6.38x106 m determine the height of the second satellite above the Earth's surface.

Homework Equations



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



So for satellite #1 we have,
T1=1.5 hr
h1=250 km above Earth's surface

For second satellite we have,
T2=7.5hr
h2=?

We know that the Earth's radius is RE=6.38x106 m

so R1=250000m+ (6.38x106m)= 6630000m

R2=RE+h2

I used (Ta/Tb)2=(Ra/Rb)3
to solve for R2=8.098x1010m

Then I did, R2-RE=h2
h2=80 970 595 km

Could someone please just verify that what I did here is correct? and if I made any mistakes if someone could please point them out to me that would be greatly appreciated! Thanks so much in advance !
 
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  • #2
dani123 said:

Homework Statement



Two satellites are orbiting around the Earth. One satellite has a period of 1.5 h and is 250 km above the Earth's surface. The second satellite has a period of 7.5 h. Using Kepler's Laws and the fact that the Earth's radius is 6.38x106 m determine the height of the second satellite above the Earth's surface.

Homework Equations



Kepler's 3rd law: (Ta/Tb)2=(Ra/Rb)3

[STRIKE]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

[/STRIKE]

The Attempt at a Solution



So for satellite #1 we have,
T1=1.5 hr
h1=250 km above Earth's surface

For second satellite we have,
T2=7.5hr
h2=?

We know that the Earth's radius is RE=6.38x106 m

so R1=250000m+ (6.38x106m)= 6630000m

R2=RE+h2

I used (Ta/Tb)2=(Ra/Rb)3
to solve for R2=8.098x1010m

Then I did, R2-RE=h2
h2=80 970 595 km

Could someone please just verify that what I did here is correct? and if I made any mistakes if someone could please point them out to me that would be greatly appreciated! Thanks so much in advance !
I crossed out a bunch of stuff you don't need for this problem.

You have definitely done something wrong! The distance from Sun to Earth is only about 1.5×108km .

The ratio of T2/T1 = 5.

The ratio of R2/R1 should be less than 5.

Did you forget to take the cube root in getting your answer?
 
  • #3
You are absolutely right, i didn't put it in my calculator correctly... so this time i got R2=8998.29 km or 8998294.4m

but from here, how do i go about getting the value for h2 which is ultimately what the question is asking for?
 
  • #4
I proceeded to do the exact thing as I did in my attempt above and got h2=2618.29 km

Does this seem reasonable?

Thanks again so much for your help!
 
  • #5
dani123 said:
You are absolutely right, i didn't put it in my calculator correctly... so this time i got R2=8998.29 km or 8998294.4m

but from here, how do i go about getting the value for h2 which is ultimately what the question is asking for?
I don't think that's right.

Show your steps in detail.
 
  • #6
I did:

R1=(250000m)+(6.38x106m)= 6 630 000m

Then I went on to find R2=Re+h

So with the equation (T1/T2)2=(R1/R2)3

(R2)3= (R1)3/(T1/T2)2

R2=(6 630 000m)3/(1.5/7.5)2= 8998294.399 m

And then I manipulated the R=Re+h equation and solved for h=2618294.399 m
 
Last edited:
  • #7
The problem asks for the height of the satellite above the Earth's surface. Aren't you calculating the distance from the Earth's center?
 
  • #8
sorry, yes your right, I just fixed it... does that seem right?
 
  • #9
does h= 2618294.399m seem like the correct answer?
 

1. What is universal gravitation?

Universal gravitation is a fundamental law of physics that describes the force of attraction between all objects with mass in the universe. It states that every object in the universe attracts every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

2. How is the height of a satellite determined using universal gravitation?

The height of a satellite can be determined using the formula h = (GMm/r^2) - R, where G is the gravitational constant, M and m are the masses of the larger and smaller object respectively, r is the distance between them, and R is the radius of the larger object (in this case, the Earth). By plugging in the appropriate values, the height of the satellite can be calculated.

3. Does the mass of the satellite affect its height in orbit?

Yes, the mass of the satellite does affect its height in orbit. The higher the mass of the satellite, the greater the gravitational force it experiences from the Earth, and therefore the higher its orbit will be.

4. How does the distance between the satellite and the Earth affect its height?

The distance between the satellite and the Earth affects its height in orbit through the inverse square law. As the distance increases, the gravitational force decreases, resulting in a larger orbit for the satellite.

5. Are there any other factors that affect the height of a satellite in orbit?

Yes, there are other factors that can affect the height of a satellite in orbit, such as the rotation of the Earth, the shape of the Earth, and the presence of other celestial bodies. These factors can lead to variations in the satellite's orbit and must be taken into account in calculations.

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