Calculating free-fall acceleration of other planets

[menglish20]

Homework Statement

Here is the text of the question:
"A satellite circles planet Roton every 2.8 h in an orbit having a radius of 1.2 X 10^7 m. If the radius of Roton is 5.0 X 10^6 m, what is the magnitude of the free-fall acceleration on the surface of Roton?"

Homework Equations

v=d/t

M= [(Ve)^2*R]/(2G)

a=GM/r^2

The Attempt at a Solution

This is presented as multiple choice, and I've been able to find the answer as 27 m/s^2 but I haven't been able to figure it out on my own.

What I tried to do is find the mass of Roton using the orbit speed.
I took the distance traveled by the satellite, 2*pi*1.2e7, and diving it by the time of one complete orbit, 2.8 hr or 10080 s. I calculated this velocity as 7479.98 m/s.

To find mass, I used the second formula I listed, and used the radius of Roton as R. I'm not sure if this is correct. I used this mass in the third formula, using the sum of Roton's radius plus the orbit radius. I think this may also be incorrect. I got an answer that was significantly different than the multiple choice answers provided. Can someone steer me in the right direction?

 

[Barakn]

To find mass, I used the second formula I listed, and used the radius of Roton as R. I'm not sure if this is correct.

No, it's not correct. The orbital radius should have been used as R.

I used this mass in the third formula, using the sum of Roton's radius plus the orbit radius. I think this may also be incorrect.

Indeed. You should have used just Roton's radius.

 

[Zula110100100]

If you know the velocity and the orbit radius you can find the acceleration, no need for mass

 

[menglish20]

If you know the velocity and the orbit radius you can find the acceleration, no need for mass

I had a guess that mass isn't needed but I couldn't figure out a way to calculate acceleration without it. Am I overthinking this? Given velocity and orbit radius, could you use:

F_c= m*a_c, so a_c=v²/r ?

If so, this is way easier than I thought.

Edit: I don't think this is correct. First, I don't even know if that equation is true, and if so, the a_c would not be the free-fall acceleration, but the centripetal acceleration that maintains the satellite in a circular path.

Edit 2: I've figured it out! You do indeed need mass. Calculate it using v=sqrt(GM/R). Then use the mass in the formula a=GM/R^2. The R in the first equation is the orbital radius and the R in the second is the radius of Roton. This may have been what Barakn was explaining but I was using the wrong formula for calculating the mass. Thanks for the help, Barakn and Zula!

 

[Barakn]

You can calculate the mass, but it's not necessary. We know that M= Ve²*R/G (you typed out the formula wrong) and a = GM/r². Substitute M into the second equation to get a = G * Ve²*R/(r²*G) = Ve²*R/r². Everything performed in one calculation in which G and M have magically disappeared.