User:Hyperquantization/Equation sandbox

Rotational Inertia

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

$F_{net}=ma$

$\tau _{net}=I\alpha$

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

$x=x_{0}+v_{0}t+{\frac {1}{2}}at^{2}$

$\theta =\theta _{0}+\omega _{0}t+{\frac {1}{2}}\alpha t^{2}$

$v_{\bot }=r\omega$

$a_{\bot }=r\alpha$

Geometry

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: ${\begin{aligned}F_{net}=ma&=mg-T\\T&=mg-ma\end{aligned}}$