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^$$\alpha$$M^$$\beta$$R^$$\gamma$$

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.

 

[tomcue]

I would first like to commend you for using dimensional analysis to find a formula for the free-fall time of an astrophysical gas ball. This is a valuable tool for understanding the relationships between different physical quantities.

Your solution is correct, as confirmed by the fact that the units on both sides of the equation are consistent. The negative exponents for each variable make sense as gravity decreases as distance increases. It is also important to note that the dimensionless constant τ represents the proportionality factor between the different physical quantities.

Regarding your question about significant figures, it is always important to use the appropriate number of significant figures in your calculations. In this case, since the constant you are using is given to 3 significant figures, you should use 6.63 in your calculation. This ensures that your final answer is not more precise than the given constant. However, if you were to use a more precise value for the constant, such as 6.6260693 x 10-34, then you should use all the significant figures in your calculation. This is because using a more precise value for the constant would imply that you have more accurate measurements for the other variables in the equation as well.