Mass of planet, given only Earth's mass?

In summary, using the given information that the speed of Saturn's moon is ten times that of Earth's moon and the mass of Earth is 6x10^24kg, we can calculate the mass of Saturn to be 6x10^26kg using the equation GmmE/r^2=mv^2/r and setting up a ratio between the two planets.
  • #1
paperclip
32
0
This is a past paper exam question:

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)



Homework Equations


GmmE/r^2=mv^2/r


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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  • #2
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):
paperclip said:

Homework Equations


GmmE/r^2=mv^2/r

Also:

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.
 
  • #3
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?
 
  • #4
Wikipedia says: Yay!
Don't forget your units.
 
  • #5
Hm... 6x10^24mg? Just kidding.

6x1-^24kg it is then!
 

1. What is the formula for calculating the mass of a planet, given only Earth's mass?

The formula for calculating the mass of a planet, given only Earth's mass, is as follows: Mass of planet = (Earth's mass * Distance between Earth and planet) / (Radius of Earth)^2

2. Does the mass of a planet depend on its distance from the Earth?

Yes, the mass of a planet is directly proportional to its distance from the Earth. This means that the farther the planet is from Earth, the smaller its mass will be, and vice versa.

3. Can the mass of a planet be calculated using only Earth's mass and radius?

Yes, the mass of a planet can be calculated using only Earth's mass and radius. However, this formula assumes that the planet has a circular orbit around the Sun and that the density of the planet is similar to Earth's.

4. How accurate is the calculation of a planet's mass using only Earth's mass?

The accuracy of the calculation depends on the accuracy of the input values, such as Earth's mass and the distance between Earth and the planet. If these values are known with a high degree of precision, the calculated mass of the planet will also be accurate.

5. Is the mass of a planet the only factor that determines its gravitational pull?

No, the mass of a planet is not the only factor that determines its gravitational pull. The distance between the planet and other objects, as well as their masses, also play a significant role in determining the strength of the gravitational force.

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