A planet has two satellite moons. Moon X has an orbital period of 2.13 days. Moon Y has an orbital period of about 3.53 days. Both moons have nearly circular orbits. Use Kepler's third law to find the distance of each satellite from the planet's center. The planet's mass is 2.0 10^26 kg.

Moon X km?
Moon Y km?

I have posted this earlier but didn't get a respond am sorry to re-post it but I really need help with this!

Physics - drwls, Sunday, February 19, 2012 at 12:27pm

The Kepler's law formula you need to use is:

T²/R³ = 4π²/(GM)

M is the mass of the planet, in kg
G is the universal constant of gravity, which is easily found online.

T is the period in seconds
Tx = 1.840*10^5 s
Ty = 3.050*10^5 s
R is the orbit radius in meters. Do the calculations of R for planets X and Y, one at a time.

Ok i tried this but i still cant get the answer right I don't understand it :(

T^2/R^3=4PI^2/GM

R= cubrt ( T^2 G*mass/4PI^2)

solve for R, knowing period T, G, Mass.
Do that for each moon.

Ok so this is what i did cubrt(2.13^2(6.67x10^-11)(2.0x10^26))/(4pi2)=R which is 1159307512 but then that's wrong so i don't know what i am doing wrong?

I apologize for the confusion. Let me explain step by step how to use Kepler's third law to find the distance of each satellite from the planet's center.

Step 1: Determine the mass of the planet. You have already been given the mass of the planet, which is 2.0 x 10^26 kg. Let's call this value M.

Step 2: Find the universal constant of gravity (G). The value of G is approximately 6.67430 x 10^-11 m^3 kg^-1 s^-2.

Step 3: Calculate the orbital period of each moon in seconds (T). You have already been given the orbital periods in days. To convert days to seconds, multiply the number of days by 24 (hours in a day), 60 (minutes in an hour), and 60 (seconds in a minute).

For Moon X:
T(X) = 2.13 days * 24 hours/day * 60 minutes/hour * 60 seconds/minute

For Moon Y:
T(Y) = 3.53 days * 24 hours/day * 60 minutes/hour * 60 seconds/minute

Step 4: Calculate the radius of the orbit for each moon (R). Rearrange Kepler's third law formula to solve for R:

R = [((T² * G * M) / (4π²))]^(1/3)

For Moon X:
R(X) = [((T(X))² * G * M) / (4π²)]^(1/3)

For Moon Y:
R(Y) = [((T(Y))² * G * M) / (4π²)]^(1/3)

Step 5: Evaluate the calculations. Substitute the values you have into the formulas and calculate the results.

Please note that the units of the answer will be in meters, which can be converted to kilometers by dividing by 1000.

Let me know if you need further assistance or if you have any additional questions.