How Do You Calculate the Density of a Planet Based on Satellite Orbit Time?

[shadowice]

Homework Statement

A satellite is in a circular orbit very close to the surface of a spherical planet. The period of the orbit is 2.50 hours. What is density of the planet? Assume that the planet has a constant density.

R= radius
G= gravational constant = 6.6742x10^-11
M = mass of
m = mass of
T = period = 9000 seconds
M = mass of earth
m = mass of planet
w= angular velocity
P = density

Homework Equations

(GmM)/R^2 = Rw^2
GPv/R^3 = W^2
w = 2pi/T
volume = 4/3pi*R^2
P= M/V

The Attempt at a Solution

i have lots of formulas and not really sure how to go about using them all
i know to start with

(GmM)/R^2 = Rw^2 then sub for w divide by r
(GmM)/R^3 = (2pi/T)^2 from here I am not sure what to do to get mass then use my M/(4/3pi*R^2)

 

[alphysicist]

Hi shadowice,

There are some erros in your equations that appear to causing some problems:

Homework Statement

A satellite is in a circular orbit very close to the surface of a spherical planet. The period of the orbit is 2.50 hours. What is density of the planet? Assume that the planet has a constant density.

R= radius
G= gravational constant = 6.6742x10^-11
M = mass of
m = mass of
T = period = 9000 seconds
M = mass of earth
m = mass of planet
w= angular velocity
P = density

Homework Equations

(GmM)/R^2 = Rw^2

This formula is not correct; you have a force on the left side and an acceleration on the right side of the equation.

GPv/R^3 = W^2
w = 2pi/T
volume = 4/3pi*R^2

This last one is also not correct; the volume is (4/3) pi R^3. Once you correct these two, do you see what to do now?

 

[shadowice]

so your saying to change this

(GmM)/R^2 = Rw^2

to

(GmM)/R^3 =w^2 to solve for a number which would be m = Mr^3*(2pi/T)^2

and set that to be the top and (4/3) pi R^3 to be the bottom and cancel out the r's and substitute numbers in. But where do i get rid of the M i see no way to remove it from the equation

[Mr^3*(2pi/T)^2]/[(4/3) pi R^3]

 

[alphysicist]

so your saying to change this

(GmM)/R^2 = Rw^2

to

(GmM)/R^3 =w^2

No, because that does not correct the equation. If you look in your book, you will see that you either have one one too many mass factors on the left side, or you are missing a mass on the right side. Once you make that small change you will almost be done with the problem.

to solve for a number which would be m = Mr^3*(2pi/T)^2

This expression cannot be right, since the units are not the same on each side of the equation.



Once you correct your first equation from this post, you will find that one of the masses cancels out and you will be left with an equation with only one mass to deal with.


After that, the goal is to put the density in the formula. You already have that:

P = M / V

so you can use this to eliminate the M in your equation.