How Do You Calculate the Mass of Mars Using Phobos' Orbit?

[Saladsamurai]

The Martian satellite Phobos travels in an approx circular orbit with r=9.4*10^6 meters and period 7h 39min. Find the mass of mars.

I am supposed to use the concept of gravitational F=centripetal force$$=m\frac{v^2}{r}$$ and the fact that $$v=\frac{2\pi r}{T}$$

so this is my attempt:

$$F_g=\frac{GMm}{r^2}=m(\frac{v^2}{r})$$

implies $$\frac{GMm}{r^2}=\frac{m4\pi^2r}{T^2}$$

implies $$M=\frac{4\pi^2r}{GT^2}$$

Should this not work? or am I just putting these numbers into my calculator wrong. I have T in seconds=275405

Text says it is 6.5*10^23 kg

Casey

ps I am more concerned with learning the method here.

 

[Kurdt]

You reasoning is fine just two small mistakes. Look at the radius in your final equation again and ask if it should be to that power, and calculate your time in seconds again.

 

[cristo]

The Martian satellite Phobos travels in an approx circular orbit with r=9.4*10^6 meters and period 7h 39min. Find the mass of mars.

I am supposed to use the concept of gravitational F=centripetal force$$=m\frac{v^2}{r}$$ and the fact that $$v=\frac{2\pi r}{T}$$

so this is my attempt:

$$F_g=\frac{GMm}{r^2}=m(\frac{v^2}{r})$$

implies $$\frac{GMm}{r^2}=\frac{m4\pi^2r}{T^2}$$

implies $$M=\frac{4\pi^2r}{GT^2}$$

You should have a factor r^3 on the RHS instead of r

Should this not work? or am I just putting these numbers into my calculator wrong. I have T in seconds=275405

Text says it is 6.5*10^23 kg

Casey

ps I am more concerned with learning the method here.

 

[Saladsamurai]

should it be r^3 ...I am messing up my algebra here...wow...
T=7h 39 min= (7*3600)+(39*60)=27540...dont know where that last 5 came from! Thanks

Casey

 

[Kurdt]

Yes. Go to the previous step and ask yourself how you move the r² term from the left to the right.

 

[cristo]

Sorry for butting in-- I was rather late! Glad you've got it though, Casey

 

[Saladsamurai]

Yes. Go to the previous step and ask yourself how you move the r² term from the left to the right.

Yeah, I'm a stooge. I treated the equals sign in this$$\frac{GMm}{r^2}=\frac{m4\pi^2r}{T^2}$$
like it was a multiplication sign!

I love making up my own rules!

Casey

 

[Saladsamurai]

Sorry for butting in-- I was rather late! Glad you've got it though, Casey

Feel free to butt in anytime

 

[Kurdt]

Yeah, I'm a stooge. I treated the equals sign in this $$\frac{GMm}{r^2}=\frac{m4\pi^2r}{T^2}$$

like it was a multiplication sign!

I love making up my own rules!

Casey

No problem. We've all made that mistake and we all will again unfortunately.

 

[dynamicsolo]

should it be r^3 ...I am messing up my algebra here...wow...

If you solve your expression for T^2, you'll see that you've derived Kepler's Third Law for Mars, so you have that a check on your algebra. (I'm assuming you've seen Kepler's Law of Celestial Motion in your course by this point.)