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Advanced Methods}Dimensional Analysis

The current author is fortunate enough to have taken a String Theory course as an undergraduate, and thus know by heart that the Planck length is \sqrt{G \hbar /c^3}. However, the problem can also be solved via dimensional analysis

(A) G\hbar c has units of \left(m^3/(kgs^2)\right) (kg m/s^2 m s) (m/s) =m^5/(s^3), which doesn't have the units of m.

(B) \left(m^3/(kgs^2)\right) (kg m/s^2 m s)^2 (m/s)^3=m^{10} kg/s^7, which doesn't have the units of m.

(C) \left(m^3/(kgs^2)\right)^2 (kg m/s^2 m s) (m/s)=m^9/(kg s), which doesn't have the units of m.

(D) \left(m^3/(kgs^2)\right)^{1/2} (kg m/s^2 m s)^2 (m/s)=\sqrt{kg}m^{4.5}/s^3, which doesn't have the units of m.

(E) This is the last one. Take it!

See below for user comments and alternate solutions! See below for user comments and alternate solutions!
Alternate Solutions
Comments
eshaghoulian
2007-10-02 04:21:34
I like barefooto's remark. It seems as if for most of the dimensional analysis problems you can pick a unit (sometimes kg, in this case meters) and see what combination will eliminate it. In this case, even if you didn't pick meters and picked kg, then all but A and E would be eliminated. You have to be careful with the meters argument, though; you could conceive of three constants with units m^2/kg, m^2kg, and m^3, stack them all next to each other, and take it to the 1/7th power at the end. So it's not necessary, in general, that one of the constants be on the bottom. But it's definitely the cleaner approach in this problem, as only choice E takes the terms to a root.
evanb
2008-06-30 13:38:24
Yes, a safer choice is to track seconds.

[G] is m^3 / kg s^2
[h] is J s which is kg m^2 / s
[c] is m/s.

We MUST eliminate seconds, so there must be a division in our answer.

Thus, the answer is E.
NEC
barefoot0
2006-11-02 14:06:48
Because all three constants have meters in the numerator you can rule out any answers that do not have at least one constant in the denominator and on in the numerator.

This gives you a shortcut to the correct answer with a bit less work.
NEC
gatboja
2005-12-08 12:48:39
trivial comments:

A: has m^6/s^4

C: has m^9/(kgs^6)
NEC

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