Finding Dimensions for Period of a Planet

[clark1089]

Homework Statement

Hi! This is from Physics for Scientists and Engineers, Vol. 6 by Tipler and Mosca.

44. Kepler's third law relates the period of a planet to its orbital radius r, the constant G in Newton's law of gravitation, and the mass of the Sun M_s . What combination of these factors gives the correct dimensions for the period of a planet?

Homework Equations

F = Gm₁m₂/r²
G = [L³/MT²]
T=C*r^a*M^b*G^c

The Attempt at a Solution

I'm trying to solve for C in the final equation given.

The big problem I'm having here comes after solving for the exponents in the third equation there...

[T] = [L]^a[M]^b[[L³/MT²]^c

Next I distribute exponents and combine the like terms...

[T] = L^a+3cM^b-cT^-2c

Okay, now I solve for C using the T¹ on the left side of the equation...

-2c = 1
c = -1/2

Now I have C, the other variables come to me... eheAHEUAEU

a+3(-1/2) = 0
b-c = 0
a = 3/2
b = -1/2

With the exponents solved for, I return to my original equation:

T=C*r^a*M^b*G^c

Only now, I sub them in.

T=C*r^3/2*M^-1/2*G^-1/2

THIS is where I have a problem! :( The answer given to this set was C = sqrt(GM)*r^3/2.

Can someone tell me what I'm doing wrong? I don't see how T gets removed or why the negative squareroots become positive while the 3/2 remains the same

ANY HELP IS APPRECIATED ! :( So stuck :(

 

[clark1089]

Made a mistake with T^(-2a).. it's actually T^(-2c)! Fixed that mistake. Sorry :X

Still need help though... :(