---
# System prepended metadata

title: Rotation of rigid bodies
tags: [Dynamics, Mechanical Mechanics]

---

# Rotation of rigid bodies
## Rotation
* 1 revolution $= 2\pi$ rad $=360^o$
$$\omega = \dot \theta$$

$$\dot\theta = d\theta/dt$$

$$\alpha = \dot\omega = \ddot\theta$$

$$\dot\omega = d\omega/dt = d^2\theta/dt^2$$

$$v = dr/dt = \omega r$$
:::success
$$a = dv/dt = \alpha r$$
:::
***
$$\theta = \theta_i+\omega t$$

$$\omega = \omega_i+\alpha t$$

$$\omega^2 = \omega_i^2+2\alpha\Delta\theta$$

$$\theta = \theta_i+\omega_i t+\frac12\alpha t^2$$
## Centripetal acceleration
$$a_r = \frac{v^2}{r}$$
:::success
$$a_r = \ddot\omega r = \omega\times(\omega\times r)$$
:::
## Momentum and Angular Momentum
$$p = mv$$

$$L = I\omega$$

$$L = mr^2\omega = rmv$$
:::success
$$L = rp$$
:::
## Rotational Energy
$$K_\omega = \frac12I\omega^2$$
