How Do You Calculate the Orbit Height of a Spacecraft Around a Planet?

[Pseudopro]

Homework Statement

I have been able to complete section a and b but c doesn't match the answer

a)The acceleration of free fall at the surface of a planet is g and the radius of the planet is R. Deduce that the period of a satellite in a very low orbit is given by T=2pi sqrt(R/g).

b)Given that g=4.5ms^-2 and R=3.4x10^6m, deduce that the orbital period of the low orbit is about 91 minutes.

c)A spacecraft in orbit around this planet has a period of 140 minutes. Deduce the height of the spacecraft from the surface of the planet. (Ans: 3.1x10^6)

Homework Equations

T=2pi sqrt(R/g)
T=2pi sqrt(R^3/GM)

The Attempt at a Solution

I can't obtain the answer 3.1x10^6 in c). When doing c), T=2pi sqrt(R/g) can't be used with g=4.5ms^-2 because period of 140 minutes means acceleration of free fall has already changed so I used T=2pi sqrt(R^3/GM):
140x60=2pi sqrt((R+h)^3/GM) {used R+h because the question wants height of spacecraft from SURFACE of planet}. Now I must find M. Using information obtained from b): 91x60=2pi sqrt((3.4x10^6)^3/GM), M=7.81x10^23. Substituting M into 140x60=2pi sqrt((R+h)^3/GM), I get R+h=4.53x10^6, h=1.13x10^6!??
I've thought this over for a very long time. Have I misunderstood the question or is my method wrong? Someone help me! Thank you!

 

[tiny-tim]

Welcome to PF!

Hi Pseudopro! Welcome to PF!

(have a pi: π and try using the X² iocn just above the Reply box )

… When doing c), T=2pi sqrt(R/g) can't be used with g=4.5ms^-2 because period of 140 minutes means acceleration of free fall has already changed so I used T=2pi sqrt(R^3/GM):

Why are you using G ?

You know g at distance R, and you know how gravity depends on distance, soooo … ?

 

[Pseudopro]

Hi Pseudopro! Welcome to PF!

(have a pi: π and try using the X² iocn just above the Reply box )


Why are you using G ?

You know g at distance R, and you know how gravity depends on distance, soooo … ?

Yes, you're right - g of 4.5 is at distance R, and gravity changes with distance. My way of calculating the new g at height h was by using g=GM/R². This is why I used G - it shouldn't be like that? This is what I did:
4.5=6.667x10^-11M/(3.4x10⁶)²
M=7.81x10²³
g_new=6.667x10^-11M(just calculated)/(R+h)²
g(R+h)²=5.20x10¹³...(1)

T=2π sqrt((R+h)/g) [cleaner way to write square root?]
140x60=2π sqrt((3.4x10⁶+h)/g)
Cleaning up: (3.4x10⁶+h)/g=1.79x10⁶...(2)

(1)x(2): (3.4x10⁶+h)³=9.31x10¹⁹
Solving: h=1.13x10⁶

If the book's answer 3.1x10^6 is correct, there is something wrong with my method. Please tell me if this (use of g=GM/R²) is incorrect. Also could you verify that my use of R+h is correct? Thank you very much for helping me

 

[tiny-tim]

Hi Pseudopro!

Your method is very long-winded.

You started with T=2π √(R³/GM) …

in other words, R is proportional to T^2/3 ……

that is all you need to know!

If the book's answer 3.1x10^6 is correct, there is something wrong with my method. Please tell me if this (use of g=GM/R²) is incorrect. Also could you verify that my use of R+h is correct? Thank you very much for helping me

I get the same R+h as you do.

I think you'll find the question has a misprint, and it should be 240 minutes, not 140.

 

[Pseudopro]

in other words, R is proportional to T^2/3 ……

that is all you need to know!

Oh! that's how you do it! I do have a knack of making simple questions complicated but still reaching the solution. Thank you very much!

 

[Pseudopro]

It's also possible they did R to T^3/2, just realized

 

[gneill]

You can dispense with G and M altogether if you wait a step or two before plugging in values.

Suppose GM = µ. Then g = µ/R², or rearranged, µ = gR².

You're given one set of values for g and R, so for another radius r,

g_r = µ/r² = gR²/r²

That's the inverse square law at work!