Circular Orbits of Satelites and Planets

1. Jun 22, 2008

.NoStyle

1. The problem statement, all variables and given/known data

A spy satellite is in circular orbit around Earth. It makes one revolution in 6.00 h.

A. How high above Earth's surface is the satellite?

B. What is the satellite's acceleration?

2. Relevant equations

I'm not sure but I believe we could possibly derive something from

SUMofF = (Gm1m2)/r^2

and with NSL, v=sqrtof (GMe/r)

where Me is the mass of the Earth 5.974e24 kg. G is the universal gravitational constant 6.67e-11N * m^2/kg^2

3. The attempt at a solution

and r is the radius of the earth plus the radius of the height. So we can find the height that the satellite is by deducting the radius of the earth (6.371e6 m) from r to find the height.

I don't know what equation to use and I don't know how I would find the acceleration.

2. Jun 22, 2008

rock.freak667

The force of attraction between the Earth and satellite,provides the force needed to keep the satellite in a circular orbit.

3. Jun 22, 2008

.NoStyle

would it be the Universal Gravitational Constant? Sorry, I'm not really clear with what you're saying. Thanks

4. Jun 22, 2008

dynamicsolo

If you set the gravitational force between Earth and satellite, given by

equal to the centripetal force acting on the satellite in its circular orbit (i.e., gravity is what provides the centripetal force), and do some algebra, you'll come up with

This is called the "circular velocity" of the spacecraft in its orbit. If you now consider that the period, T, of the satellite is the time it takes for the spacecraft to complete the circular orbit of radius R at the speed v given above, you can do some more algebra to find a relation between T and R (you will, in fact, have found a version of Kepler's Third Law!).

The gravitational force equation will give you the satellite's acceleration in its orbit, if you keep in mind Newton's Second Law (F = ...?). [Since what they are asking for is also the centripetal acceleration of the satellite, you can use that equation and the orbital speed you found. You should -- you'd better! -- get the same answer either way...]

5. Jun 22, 2008

.NoStyle

Hi again dynamic solo, thanks for all the help lately. Before I go further, r in the "circular velocity" equation is the radius of earth? thanks

6. Jun 22, 2008

rock.freak667

r is the distance between the earth and the satellite.

7. Jun 22, 2008

.NoStyle

Thank you. So for V, that would be 2pi/6h?

8. Jun 22, 2008

dynamicsolo

What is the distance around the circular orbit? That is the distance you are covering in those six hours.

9. Jun 22, 2008

.NoStyle

The distance is 2pi, isn't that right? so 2pi/6h?

10. Jun 22, 2008

dynamicsolo

The circumference of a circle is given by what formula?

(The number pi has no units, so 2·pi can't be the length of anything...)

11. Jun 22, 2008

.NoStyle

hi dynamic,

but we don't know the radius. So I have

solving for r I get:

r ^1.5 = sqrt of (GMe) * 6 all over 2pi

oh god, what am I doing?

12. Jun 22, 2008

dynamicsolo

OK, the circular speed is given by the distance around the circular orbit (circumference) divided by the time required to do so (period), so you get

But watch your algebra! You have r on one side and a square root of r in the denominator of the other side. Square both sides (also convert the period into seconds, since we'll need to work in SI metric) to get

[ (2·pi) · r ]^2 / (6 hours converted to seconds)^2
= G · Me / r .

Now solve this for r , which is the radius of the orbit. (Actually, what you had is close, but r^1.5 will be more of a nuisance to work with...)

Last edited: Jun 22, 2008
13. Jun 22, 2008

.NoStyle

hey dynamic, once again, thanks a lot for the help.

I got it right I think!

1.039E7 meters!!!

hehehe

14. Jun 22, 2008

dynamicsolo

15. Jun 23, 2008

.NoStyle

hmmm i needed help with that,

I got 8.7E-1 meters/second

that's wrong huh?

thanks

16. Jun 23, 2008

dynamicsolo

Well, acceleration has units of meters/(second^2), so something didn't come out right...

You should probably show how you calculated this. In the meantime, here are two approaches you can try (there's a third, but I'll describe it afterwards):

You know that the force on the satellite is F = ma , so you could compute either

1) the gravitational force between the satellite and the Earth, set that equal to ma , and solve for a

or

2) find the centripetal force on the satellite (you have the radius of the orbit and the speed of the satellite along it) and set that equal to ma, and solve for a .

Notice that you don't need to know the satellite's mass m, since it will divide out (which tells us that any satellite which is small in mass compared to Earth would behave the same way).

17. Jun 23, 2008

.NoStyle

Hi Dynamicsolo,

yes it is 8.7E-1 m/s^2

this is what I did.

I used the formula asubr = omegasquared*radius

i found the requency and plugged that into the equation

arrived with that answer. The problem is, I don't know if I did it 100% correctly.

I'll try your method and report back. Thank you greatly

18. Jun 23, 2008

.NoStyle

I'm a bit confused with steps 1 and 2 since there are many names for the same thing. First off, what do you mean "gravitational force" is that the same thing as Fg = Gm1m2/r^2??

and with what you're saying, we don't need to know the mass of the satellite therefore:

Fg= GMe/r^2 ???

and then plugging in the universal constant and multiplying it with the mass of earth and dividing everything by the radius of the orbit squared will give us the gravitational force? Thanks

edit -

and since you said

gravitational force = m * a

what is the value of m? which mass is it? thanks

19. Jun 23, 2008

dynamicsolo

Ah, that form for the centripetal acceleration will work too. It also looks like you found the angular frequency of the satellite on its orbit correctly. But, did you use the radius of the orbit, or the altitude...?

20. Jun 23, 2008

dynamicsolo

That is the equation for the gravitational force between two masses.

You left out one of the masses this time. The gravitational force between the Earth (mass: Me) and the satellite (call its mass m) will be

F_g = G · Me · m / (r^2) ,

which will equal the centripetal force on the satellite (since the gravitational force is what is providing that force)

F_g = F_c = m · a_c ,

the centripetal acceleration being what you are solving for. The mass m appears on both sides of the equation, so it divides out (meaning that, for satellites with masses that are small compared to Earth, the mass doesn't matter -- we must modify this approach to deal with, say, the Moon).