---
# System prepended metadata

title: Oscillation
tags: [mechanics, SHM]

---

# Oscillation
![sin wave](https://hackmd.io/_uploads/ByVm4GD4lg.png)

* equation Displacement versus time : $x(t)$
* amplitude : $x_m$
* angular frequency : $\omega$
* Phase : $\phi$

$$x(t) = x_mcos(𝜔t+\phi)$$

* $x_{max} = x_m$
* $x_{min} = -x_m$
## Cycle VS Frequency
$$T = \frac{1}{f}$$
## Place
$$x(t) = x_mcos(𝜔t+\phi)$$
* if $\phi = 0$ , then $x(t) = x_mcos(𝜔t)$
$$x(t) = x_mcos(𝜔t) = x_mcos𝜔(t+T)$$

$$𝜔(t+T) = 𝜔t+2\pi$$

$$𝜔T = 2\pi$$
:::success
$$𝜔 = \frac{2\pi}{T} = 2\pi f$$
:::
## Velocity
$$v(t) = \frac{dx(t)}{dt} = \frac{d}{dt}(x_mcos(𝜔t+\phi))$$
:::success
$$v(t) = -𝜔x_msin(𝜔t+\phi)$$
:::
## Acceleratiion
$$a(t) = \frac{dv(t)}{dt} = \frac{d}{dt}(-𝜔x_msin(𝜔t+\phi))$$

$$a(t) = -𝜔^2x_mcos(𝜔t+\phi)$$
:::success
$$a(t) = -𝜔^2x(t)$$
:::
## Graph
![oscillation](https://hackmd.io/_uploads/HJDvnXnSJe.png)
