How do you find what the mass on the spring is if you know the period and force constant of the harmonic oscillator?

Start with the equation for the period

`T = 2pisqrt(m/k)" "`, where

`T` - the period of oscillation;
`m` - the mass of the oscillating object;
`k` - a constant of proportionality for a mass on a spring;

You need to solve this equation for `m`, so start by squaring both sides of the equation

`T^2 = (2pi * sqrt(m/k))^2`

`T^2 = (2pi)^2 * (sqrt(m/k))^2`

`T^2 = 4pi^2 * m/k`

Now all you have to do is isolate `m` on one side of the equation

`T^2 * k = 4pi^2 * m`

`m = (T^2 * k)/(4pi^2) = color(green)(k * T^2/(4pi^2))`

Let's say we started from `omega = sqrt(k/m)`. It's a bit different but a similar approach.

If we examine the equation

`y = Asin(ntheta + phi) + k`

If `n` was doubled, the frequency would be doubled, but the period would be halved. So, we know that `omega prop 1/T`.

If `omega` is `2pi "rad/s"`, the period `T` is `"1 s"`, so to create the equality between the two variables, we match up the units by multiplying `1/T` by `2pi "rad"` to get `color(green)(omega = (2pi)/T)`.

`omega = (2pi)/T = sqrt(k/m)`

Square both sides:
`(4pi^2)/T^2 = k/m`

Reciprocate both sides and then multiply by `k`:
`color(blue)(m = (kT^2)/(4pi^2))`