User:Hyperquantization/Equation sandbox

General equations
$$I = MR^2$$

$$I = \frac{1}{2}MR^2$$

$$I = \frac{1}{12}ML^2$$

Specific equations
$$I_{pulley} = \frac{1}{2}MR^2$$

$$I_{ring} = MR^2$$

$$I_{brass} = MR^2$$

$$I_{disk} = \frac{1}{2}MR^2$$

$$I_{rod} = \frac{1}{12}ML^2$$

Motion
$$F_{net} = ma$$

$$\tau_{net} = I\alpha$$

$$\vec \tau = \vec r \times \vec F$$

$$x = x_0 + v_0t + \frac{1}{2}at^2$$

$$\theta = \theta_0 + \omega_0t + \frac{1}{2}\alpha t^2$$

$$v_\bot = r\omega$$

$$a_\bot = r\alpha$$

Superquadric
$$\left|\frac{x}{a}\right|^\alpha + \left|\frac{y}{b}\right|^\beta + \left|\frac{z}{c}\right|^\gamma = 1$$

Other/Derived
Tension (T) in the string: $$\begin{align}F_{net} = ma & = mg - T \\ T & = mg - ma\end{align}$$