# How does mass affect angular velocity?

For a freely rotating rigid body

$\omega \setminus \propto \frac{1}{m}$

#### Explanation:

For a freely rotating rigid body having mass $m$, radius of gyration $k$ & an angular velocity $\omega$, net torque ${T}_{\setminus \textrm{\ne t}}$ on the body must be zero

${T}_{\setminus \textrm{\ne t}} = 0$

$I \left(\setminus \frac{d \setminus \omega}{\mathrm{dt}}\right) = 0$

$\setminus \frac{d \left(I \setminus \omega\right)}{\mathrm{dt}} = 0$

$I \setminus \omega = \setminus \textrm{c o n s t}$

$m {k}^{2} \setminus \omega = \setminus \textrm{c o n s t}$

$\setminus \omega = \setminus \frac{\setminus \textrm{c o n s t}}{m {k}^{2}}$

$\omega \setminus \propto \frac{1}{m}$

thus when the mass $m$ of a freely rotating rigid body increases its angular velocity $\setminus \omega$ decreases & vice-versa
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