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.
τ = kG[tex]\alpha[/tex]M[tex]\beta[/tex]R[tex]\gamma[/tex]
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)[tex]\alpha[/tex](g)[tex]\beta[/tex](cm)[tex]\gamma[/tex]
Rearranging I found easily that [tex]\alpha[/tex] = -1/2, [tex]\beta[/tex] = -1/2 and [tex]\gamma[/tex] = -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 text book 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,