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))#