Question

What is the proof of T=2π√m/k?Without calculus.

Answer 1

Answer:

You can get a proportionality using one of the most basic techniques in science: dimensional analysis. What you do is look at the dimensions of the various quantities in the problem and try to see how they can fit together. Here you want the period, which is a time. I'll write [T] to indicate that it has dimensions of time. What could it depend on? If we're ignoring friction (which is often reasonable) the only things it could depend on are:

The mass of the pendulum m, with dimensions of mass [M]

The length of the pendulum ℓ, with dimensions of length [L]

The acceleration of gravity g, with dimensions [L/T2]

You want to put this together in a way that gives you a time. The most general thing you can write down is

maℓbgc,

for some numbers a,b,c. The dimensions of this are

[MaLb+cT−2c],

which you can see by putting in the dimensions and multiplying it out. Equating the dimension to [T1] we find:

ab+c−2c===0,0,1.

From which you find a=0, b=−c=1/2. So we know that

T=kℓg−−√,

where k is some pure number (no dimensions) which we can't work out this way. It turns out that k=2π but dimensional analysis can only take you so far. I don't know of a way to get k without using at least a little bit of calculus. :)

Note: More generally the period can and does depend on the magnitude of the swing, given by θmax, the largest angle of deflection from the vertical. Angles are dimensionless quantities, so really we can only write