Solving for Moon B's Orbital Period: A Homework Challenge

[Devin Longo]

Homework Statement

Two moons orbit a planet in nearly circular orbits. Moon A has orbital radius r, and moon B has orbital radius 5r. Moon A takes 50 days to complete one orbit. How long does it take moon B to complete and orbit?

Homework Equations

Kepler's Third Law

The Attempt at a Solution

I know we're supposed to attempt it, but I'm not even sure I'm heading in the right direction! Can anybody give me some insight?

 

[mgb_phys]

have you tried looking up kepler's 3rd law /

 

[kerimek]

Sure, I'd be happy to help you out! First, let's review Kepler's Third Law, which states that the square of a planet's orbital period is directly proportional to the cube of its semi-major axis (in other words, the radius of its orbit). Mathematically, this can be written as T^2 = k*a^3, where T is the orbital period, a is the semi-major axis, and k is a constant.

Now, let's apply this to our problem. We know that moon A has an orbital period of 50 days and an orbital radius of r. So, we can write the equation T^2 = k*r^3 for moon A.

For moon B, we know that its orbital radius is 5r, so we can write the equation T^2 = k*(5r)^3.

Since we're solving for the orbital period of moon B, we can set these two equations equal to each other and solve for T. This gives us:

T^2 = k*r^3 = k*(5r)^3

T^2 = k*r^3 = 125k*r^3

T^2 = 125k*r^3

T = √(125k*r^3)

Now, we can substitute in the values we know for k and r (assuming we are using the same units for both moons). This gives us:

T = √(125*50^3) = 250 days

Therefore, it takes moon B 250 days to complete one orbit. I hope this helps and good luck with the rest of your homework!