Dimensional Analysis of Free-Fall Time of Astrophysical Gas Ball

[CaptainEvil]

Homework Statement

Use dimensional analysis to find a formula for the free-fall time τ of an astrophysical ball of gas. In this system, the only force is gravity, therefore the only quantities are Newton’s constant, G, and the mass and the radius of the sphere, M and R respectively.

Homework Equations

τ = kG^\alphaM^\betaR^\gamma

where τ is a dimensionless constant.

The Attempt at a Solution

I am using cgs units, and want to satisfy the equation dimensionally.
on the left side we have (s) obviously, and on the right side I have (cm³g^-1s^-2)^\alpha(g)^\beta(cm)^\gamma

Rearranging I found easily that \alpha = -1/2, \beta = -1/2 and \gamma = -3/2

Unless I am missing something, that's the answer, but it doesn't look right to me. Can anyone confirm this?

Also, I had a question regarding sig figs of fundamental constants. If I am asked to use physical constants to 3 significant digits - and my high-end physics textbook tells me a constant like Planck's constant is 6.6260693 x 10^-34, will I use 6.62, or round up to 6.63 for 3 significant digits. Thanks for the hasty reply,

CaptainEvil

 

[kuruman]

Your answer is OK. To convince yourself, square both sides of your equation, take the ratio R³/T² and compare against Kepler's Third Law.

6.626 is closer to 6.630 than to 6.620. Round up.