Find mass of mars given period of moon orbit and radius of orbit

[buttermellow]

Homework Statement

Given data: A moon of Mars orbits with a period of 459 minutes. The radius of the moon's orbit is 9.4x10^6 m. What is the mass of mars?

Homework Equations

The Attempt at a Solution

The only relevant equation I could find was F_{mars on moon}= (G*m₁*m₂)/(r²)

I have the radius, but to solve I would need the mass of the moon and the gravitational force. Could I use F=mv²/r ? Or how about v= $$\sqrt{}GM/r$$ ?

 

[buttermellow]

Alright, I actually figured it out myself.

v= $$\sqrt{}Gm/r$$ and v=$$\omega$$*r

so the mass of Mars is 6.5*10^23 kg!

 

[tomcue]

I would approach this problem by first acknowledging that the given data is not sufficient to directly calculate the mass of Mars. We would need additional information such as the mass of the moon and the gravitational force between the moon and Mars. However, we can make some assumptions and use some equations to estimate the mass of Mars.

Firstly, we can assume that the moon's orbit is circular, which means that the centripetal force acting on the moon is equal to the gravitational force between the moon and Mars. This can be represented by the equation F=mv^2/r, where m is the mass of the moon, v is the velocity of the moon, and r is the radius of the moon's orbit.

We can also use the equation for the period of an orbit, T=2π√(r^3/GM), where T is the period, r is the radius of the orbit, G is the gravitational constant, and M is the mass of the planet.

By substituting the given values into these equations, we can create a system of equations and solve for the mass of Mars. However, since we do not have the mass of the moon, we can make another assumption and use the mass ratio between the moon and Mars. This can be calculated by using the equation m2/m1= r2/r1, where m2 and r2 represent the mass and radius of the moon, and m1 and r1 represent the mass and radius of Mars.

By substituting this ratio into our system of equations, we can solve for the mass of Mars. However, it is important to note that this will only give us an estimate and the actual mass of Mars may differ slightly from this calculated value. Additionally, this calculation assumes that the moon's orbit is not affected by any other gravitational forces, which may not be entirely accurate.

In conclusion, while we can use some equations and assumptions to estimate the mass of Mars, we would need more information to accurately calculate it. This highlights the importance of having complete and accurate data in scientific research.