A periodic motion has the following characteristics:
- Period $T$ in s
- frequency $f=\frac{1}{T}$ in s$^{-1}$ or Hz
- angular frequenct $\omega=\frac{2\pi}{T}$ in rad s$^{-1}$
- an amplitude $A$ in m
- a displacement function $x(t)=A\cos(\omega t)$
$$ F=ma\quad\Rightarrow\quad m\frac{\text{d}^2x}{\text{d}t^2}=-kx $$
$$ \omega^2=\frac{k}{m} \quad \text{and}\quad \ddot{x}=-\omega^2 x $$
$$ x(t)=A\sin(\omega t+\Phi)=a\cos(\omega t)+b\sin(\omega t) $$
The basketball can be grabbed to change the values of $A$ and $\Phi$, then $\omega$ can be changed using the top slider
$$ \begin{aligned}m\ddot{x}=m\dot{v}&=mv\frac{\text{d}v}{\text{d}x}=-kx\\ \Rightarrow \quad \text{d}\left(\frac{1}{2}mv^2\right)&=\text{d}\left(-\frac{1}{2}kx^2\right) \\ \Rightarrow\quad\underbrace{\frac{1}{2}mv^2}{E_k}+\underbrace{\frac{1}{2}kx^2}{E_p}&=\underbrace{\text{Constant}}_{E_m} \end{aligned} $$
$$ E_m=\frac{1}{2}mA^2\omega^2\sin^2(\omega t+\Phi)+\frac{1}{2}kA^2\cos^2(\omega t+\Phi)=\frac{1}{2}kA^2 \\ \text{so: }\quad E_m\propto A^2 $$
<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/2ba5ecef-fd4f-42ec-abf8-b7e1a677c5cc/Simple_pendulum.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/2ba5ecef-fd4f-42ec-abf8-b7e1a677c5cc/Simple_pendulum.png" width="40px" /> Simple pendulum
</aside>
$$ \begin{aligned} F&=-mg \sin(\theta)\\ ml\ddot{\theta}&=-mg\sin(\theta)\\ \text{for }\theta\approx0 \quad\ddot\theta&=-\frac{g}{l}\theta\\ \text{sub }x\Rightarrow\theta&=\theta_0\cos(\omega t+\Phi) \end{aligned}
$$
<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/5d81a54b-ac24-4c8b-a60b-7132f0f13b9d/Floating_object.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/5d81a54b-ac24-4c8b-a60b-7132f0f13b9d/Floating_object.png" width="40px" /> Floating object
</aside>
$$ \text{Using archimedes principle:}\quad F=-\rho Axg\Rightarrow\omega^2=\frac{\rho Ag}{m} $$
<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/dc5b8cd8-7786-4bad-8c7b-f1df79faa32c/AC_electric_circuit.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/dc5b8cd8-7786-4bad-8c7b-f1df79faa32c/AC_electric_circuit.png" width="40px" /> AC electric circuit
</aside>
$$ \begin{aligned} \bullet&\;V_L=L\frac{\text{d}I}{\text{d}t}\;\small\text{across the inductor} \\ \bullet&\;V_C=\frac{q}{C}\;\small{\text{across the capacitor, }q\text{ is the charge}} \end{aligned} $$
$$ L\frac{\text{d}I}{\text{d}t}+\frac{q}{C}=0 $$
$$ \frac{\text{d}}{\text{d}t}\left[L\frac{\text{d}I}{\text{d}t}+\frac{q}{C}=0\right]\Rightarrow\frac{\text{d}I}{\text{d}t^2}+\frac{I}{C}=0\Rightarrow\omega=\frac{1}{\sqrt{LC}} $$
