Damon Vazquez

2022-09-01

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

Cameron Wallace

$T=2\pi \sqrt{\frac{m}{k}}$, 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
${T}^{2}=\left(2\pi \ast \sqrt{\frac{m}{k}}{\right)}^{2}$
${T}^{2}=\left(2\pi {\right)}^{2}\ast \left(\sqrt{\frac{m}{k}}{\right)}^{2}$
${T}^{2}=4{\pi }^{2}\ast \frac{m}{k}$
Now all you have to do is isolate m on one side of the equation
${T}^{2}\ast k=4{\pi }^{2}\ast m$
$m=\frac{{T}^{2}\ast k}{4{\pi }^{2}}=k\ast \frac{{T}^{2}}{4{\pi }^{2}}$

aurelegena

Let's say we started from $\omega =\sqrt{\frac{k}{m}}$. It's a bit different but a similar approach.
If we examine the equation
$y=A\mathrm{sin}\left(n\theta +\varphi \right)+k$
If n was doubled, the frequency would be doubled, but the period would be halved. So, we know that $\omega \propto \frac{1}{T}$
If $\omega$ is $2\pi rad/s$, the period T is 1 s, so to create the equality between the two variables, we match up the units by multiplying $\frac{1}{T}$ by $2\pi rad$ to get $\omega =\frac{2\pi }{T}$
$\omega =\frac{2\pi }{T}=\sqrt{\frac{k}{m}}$
Square both sides:
$\frac{4{\pi }^{2}}{{T}^{2}}=\frac{k}{m}$
Reciprocate both sides and then multiply by k:
$m=\frac{k{T}^{2}}{4{\pi }^{2}}$

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