Two satellites in orbit around Jupiter

[magusxion]

Homework Statement

Two satellites are in circular orbits around Jupiter. One, with orbital radius r, makes one revolution every 16 h. The other satellite has orbital radius 4.0r. How long does the second satellite take to make one revolution around Jupiter?

A. 130 hours
B. 120 h
C. 140 h
D. 110 h
E. 90 h

Homework Equations

v = sqrt( GMj / r )

T = (2*pi*r) / v

G = 6.673*10^-11

Mj = 1.90*10^27

The Attempt at a Solution

I'm assuming that this is just an algebra problem at this point, but I'm having trouble finding r.

T = ( 2*pi*r) / sqrt( GMj / r )

r = [(GMjT^2) / (4pi^2)]^(1/3)

When I solve for r and then plug r back into the equation for T I do not get 16, which means I'm not doing something right here.

Do I need to convert the 16 hours / revolution into another unit before solving for r?

Any pointers?

 

[gneill]

According to Kepler's law the square of the period varies as the cube of the orbit radius.

$$T^2 \propto R^3$$

This implies that for any given set of orbits around a common central body, that

$$\frac{T^2}{R^3} = Constant$$

You should be able to use this fact to determine the "mystery" orbital period.

 

[magusxion]

Thanks, that did help a lot.

It seems like the method I was trying to work through would also work but is much longer.

It also doesn't help that when I do the calculation with either method I still do not get a value the exactly matches one of the answer choices, but is very close to answer A ( which did turn out to be correct ).