Dimensional Reasoning

Satya:

Chandra…I forgot the formula for the period of a pendulum.

Chandra:

Forgetting formulas is normal.

Satya:

I remember something $2\pi$ …. $g$ ….. $l$ … umm…but not exactly

Chandra:

Then don’t memorize the formula. Let dimensions guide you.

Satya:

Dimensions? How?

Chandra:

The period is time period. Time has dimension of [T], length has [L], and g has [L][T]^-2. Now combine them so that the final result has dimension [T].

Satya:

I got it

$T \propto \sqrt{\frac{l}{g}}$

Chandra:

Exactly. No memorization, just logic.

Satya:

Wow.. so I can rebuild and verify formulas!

Chandra:

Correct. Now tell me quickly,

$v=u+at^2$

right?

Satya: Yeah.. ahh.. wait a minute

Nooo! My brain memorized, but dimensions caught the error.

$v$ and $u$ are [L][T]^-1 ,

$a t^{2}$ is [L]

Therefore It must be

$v=u+at$

Now I got a spell check for equation.

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