Universal gravitation 1-determine height of satellite

[dani123]

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: (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

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 R_E=6.38x10⁶ m

so R₁=250000m+ (6.38x10⁶m)= 6630000m

R₂=R_E+h₂

I used (T_a/T_b)²=(R_a/R_b)³
to solve for R₂=8.098x10¹⁰m

Then I did, R₂-R_E=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 !

 

[SammyS]

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: (T_a/T_b)²=(R_a/R_b)³

[STRIKE]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

[/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 R_E=6.38x10⁶ m

so R₁=250000m+ (6.38x10⁶m)= 6630000m

R₂=R_E+h₂

I used (T_a/T_b)²=(R_a/R_b)³
to solve for R₂=8.098x10¹⁰m

Then I did, R₂-R_E=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?

 

[dani123]

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?

 

[dani123]

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!

 

[SammyS]

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.

 

[dani123]

I did:

R₁=(250000m)+(6.38x10⁶m)= 6 630 000m

Then I went on to find R₂=R_e+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=R_e+h equation and solved for h=2618294.399 m

 

[HallsofIvy]

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?

 

[dani123]

sorry, yes your right, I just fixed it... does that seem right?

 

[dani123]

does h= 2618294.399m seem like the correct answer?