# Mass of planet, given only Earth's mass?

#### paperclip

This is a past paper exam question:

1. Homework Statement
One of the moons of Saturn is in an orbit which has approximately the same radius as that of the Earth's moon. Given that the speed of the saturn moon is ten times the speed of the Earth's moon, calculate a value for the mass of Saturn.
(Mass of the Earth=6x10^24kg)

2. Homework Equations
GmmE/r^2=mv^2/r

3. The Attempt at a Solution
Mass of earth=6x10^24
r satmoon=r earthmoon
v satmoon=v earthmoon/10

I did try this, but it's obviously wrong:
tau^2=4pi^2r^3/Gme

G=(6.67x10^-11) me=(6x120^24)

period of revolution of the moon=24hrs

hence (24)^2=4pir^3/4x10^14

576=4pi r^3
144=pir^3

even without going further, you can tell that it's going to be faaar too small to represent the radius of the moon.

Without being given anything else-- do you need to memorise the radius of the moon's orbit for the exam? I can't really see any other way! I'm stumped...please help!

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#### Doc Al

Mentor
You don't need to know the radius of the moon's orbit. Since both Earth and Saturn moons have the same orbital radius, treat that radius as a constant.

How does planetary mass affect the moon's speed? Hint: Set up a ratio, using this equation (generalize it for any planet, of course):
2. Homework Equations
GmmE/r^2=mv^2/r
Also:
3. The Attempt at a Solution
Mass of earth=6x10^24
r satmoon=r earthmoon
v satmoon=v earthmoon/10
You seem to have that last relationship reversed--Saturn's moon is 10x faster, not slower, than Earth's moon.

#### paperclip

Thanks so much for your help. I went back and tried it again:
if v=(sqrt)Gme/r, then set up ratio:

10(sqrtGmearth/r)= sqrt Gmsaturn

Remove r as it is a a constant
This gives us
10(sqrt G.mearth)=G.msaturn Remove G as it is another constant.
10(sqrt mearth)=sqrt msaturn
10 (sqrt 6x10^24=sqrt msaturn
2.45x10^13=sqrt msaturn (square both sides)
(2.45x10^13)^2=msaturn=6x10^26

therefore mass of saturn=6x10^26. Yay or nay?

#### tony873004

Gold Member
Wikipedia says: Yay!