How Far is an Object Orbiting Earth if Its Period is 32 Days?

[Digdug12]

Homework Statement

An object of mass 7.1x1023 circles the Earth and is attracted to it with a force whose magnitude is given be Gmem/r2. If the period of rotation is 32 days, what is the distance from the Earth to the object? Here G=6.67x10-11 Nm2/kg2, me=6x1024 kg.

Homework Equations

F_g=G*m*M_e/r²
T=2pi/Omega

The Attempt at a Solution

I used the first equation to isolate r, the distance from the Earth to the object to get sqrt((G(m*M_e)/F_g)=r
But this is where I get stuck, i don't see how to find the other unknown, F_g I'm guessing it has to do something with the period of 23 days, but I cannot connect the two in my head.

 

[Doc Al]

Hint: Apply Newton's 2nd law. How is the object accelerating?

 

[Digdug12]

I set it up so that Fg=Ma_c, since A_c=V²/r, but the radius is still in the equation. If substituted it back in i would have another unknown, velocity. What am i missing here? is there another force?
edit:
Should i use v=r*omega, Period=2pi/omega, and substitute v=r*(28/2pi) into the Fg=M(V^2/r)?

edit2:
i got the final equation GMeM/M(28/2pi)^2=r but its off by roughly 1x10^2., is there a problem with the period? I see that i used 28 days instead of 32 but i tried it with 32 also and it was still off, do i need to convert days into seconds or something?

 

[Doc Al]

edit:
Should i use v=r*omega, Period=2pi/omega,

Good.

and substitute v=r*(28/2pi) into the Fg=M(V^2/r)?

You messed that step up a bit. Redo.

 

[Digdug12]

ok so i setup it up to look like this:
F_g=m*r*(32$$/2pi$$)²
Then i substituted it back into the original equation and i got:
G*M*M_e/M*(32$$/2pi$$)²=r
and this comes out to 1.87e10
the correct answer is supposed to be 4.26e8

 

[Doc Al]

ok so i setup it up to look like this:
F_g=m*r*(32$$/2pi$$)²

Still not right. For some reason, you are inverting things. v = r*(2pi/T), not r*(T/2pi). Redo your equation for v from post #3.

 

[Digdug12]

ahh thanks for catching that Doc. Al :D I'm sorry i still haven't solved it, I am really getting frustrated haha... i made the change in the equation but my answer is still off, is the mass in F_g=M*a_c the mass of the object, right?

 

[Doc Al]

is the mass in F_g=M*a_c the mass of the object, right?

Right.

Show me your final equation.

 

[Digdug12]

for the first few tries i canceled one r on each side of the equation, it came out to be
r=G*M*M_e/M*(2pi/32)^2

then i realized i was cancelling a r with a 1/r, so i carried the r^2 over and it became r^3, so i took the cube root of the answer, tried cancelling the M's, and still nothing.

 

[Doc Al]

The M's will cancel and you will need to take a cube root. What's the final version of your equation for r³?

Be sure to convert the period from days to seconds! (D'oh!)

 

[Digdug12]

oooo, sweet. thanks, I got it now :D thanks a bunch