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http://ocw.mit.edu/courses/physics/8-01sc-physics-i-classical-mechanics-fall-2010/

I'm already frustrated in the first module. One major lesson involves guessing a formula based on making the dimensions come out right. One question asks for a formula for the period T of a swinging pendulum. We are told that we should think of some physical quantities on which T might depend. They suggest m, the mass of the bob at the end of the pendulum, θ, the angular amplitude, L, the length of the pendulum, and g, the gravitational acceleration. They say that T cannot depend on m, because none of the other physical quantities have mass units that could cancel out the mass unit of the bob in a formula in order to leave just time units for the period T. Then they talk about how to combine the given quantities to get time units out, and come to the conclusion that T is proportional to the square root of L / g (up to a multiplicative function of θ since the angle is dimensionless).

This is all logical to me except for one thing. Why do the dimensions on each side of the equation have to agree in the first place? For example, the notes also mention Hooke's Law for springs: F = -kx. The dimensions of F are M L / T^2 (M = mass, L = length, T = time) but those for x are just L. The units don't match. They get around this by saying that the dimensions of the spring constant k are

M / T^2. This makes the dimensions on each side of the equation match up. Well, why can't the formula for the period of the pendulum also have some constant with units designed to make the dimensions of each side of the equation match up?

(I know that the formula from the period can be derived and it is indeed proportional to the square root of L / g and it does not depend on m.)

Thanks for any help.