Question

how do you find the dimensions of alpha??

Answer 1

The argument is that the time it takes the apple to fall must depend on the height, because that's common sense. But we don't know how tt varies with hh. For example we could have:

t∝ht∝h

or

t∝h2t∝h2

or

t∝h12t∝h12

and so on. So the lecturer is saying, because we don't know how t depends on h let's just put:

t∝hαt∝hα

and then we'll use dimension analysis to calculate αα. Likewise the lecturer is assuming that mass and gravity are also important, and again we don't know how the time depends on them, so he's putting them in to the power ββ and γγ and then he'll calculate what ββ and γγare. That's why you get:

t=hα×mβ×gγt=hα×mβ×gγ

The cunning trick is that hh has units of length (LL), mm has units of mass (MM) and gg (an acceleration) has units of length per second22 (LT−2LT−2), and of course time has units of time (TT). If we put these into our equation we get:

T=Lα×Mβ×(LT−2)γT=Lα×Mβ×(LT−2)γ

The only way this can be true is if β=0β=0, γ=−12γ=−12 and α=12α=12 i.e.

t∝hg−−√