Average Angular Velocity:

$\omega_{av} = \frac{\Delta \theta}{\Delta t}$

Instantaneous Angular Velocity:

$\omega = \frac{d\theta}{dt}$

Average angular acceleration: $\alpha_{av} = \frac{\Delta \omega}{\Delta t}$

Instantaneous Angular Acceleration:

$\alpha = \frac{d\omega}{dt} = \omega \frac{d\omega}{d\theta}$

Relation Between Speed And Angular Velocity:

$v = r\omega$

and $\vec{v} = \vec{\omega} \times \vec{r}$

Tangential Acceleration (Rate Of Change Of Speed):

$a_t = r \frac{d\omega}{dt} = \omega \frac{dr}{dt}$

Radial Or Normal Or Centripetal Acceleration:

$a_r = \frac{v^2}{r} = \omega^2 r$

Total Acceleration:

$\vec{a} = \vec{a}_t + \vec{a}_r \Rightarrow a = \sqrt{a_t^2 + a_r^2}$

Angular Acceleration:

$\vec{\alpha} = \frac{d\vec{\omega}}{dt} \quad \text{(Non-uniform circular motion)}$

Radius Of Curvature:

$R = \frac{v^2}{a_{\perp}} = \frac{mv^2}{F_{\perp}}$

Normal Reaction Of Road On A Concave Bridge:

$N=m g \cos \theta+\frac{m v^2}{r}$

Normal Reaction On A Convex Bridge:

$N=m g \cos \theta-\frac{m v^2}{r}$

Skidding Of Vehicle On A Level Road:

$v_{\text{safe}} \leq \sqrt{\mu gr}$

Skidding Of An Object On A Rotating Platform:

$\omega_{\max} = \sqrt{\frac{\mu g}{r}}$

Bending Of Cyclist:

$\tan \theta = \frac{v^2}{rg}$

Banking Of Road Without Friction:

$\tan \theta = \frac{v^2}{rg}$

Banking Of Road With Friction:

$\frac{v^2}{rg} = \frac{\mu + \tan \theta}{1 - \mu \tan \theta}$

Maximum And Minimum Safe Speed On A Banked Frictional Road:

$V_{\max} = \left[\frac{rg(\mu + \tan \theta)}{(1 -\mu \tan \theta)}\right]^{1/2}$

$V_{\min} = \left[\frac{rg(\tan \theta -\mu)}{(1 + \mu \tan \theta)}\right]^{1/2}$

Centrifugal Force (Pseudo Force):

$f = m\omega^2 r$

It acts outwards when the particle itself is taken as a frame.

Effect Of Earth’s Rotation On Apparent Weight:

$N = mg - mR\omega^2 \cos^2 \theta,$

where $\theta$ is the latitude at a place.

Vertical Circle:

Various quantities for a critical condition in a vertical loop at different positions.

Conical pendulum:

$T \cos \theta = mg$

$T \sin \theta = m\omega^2 r$

Time period: $T = \sqrt{\frac{2\pi L \cos \theta}{g}}$

Relations among angular variables:

Initial angular velocity: $\omega_0$

$\omega = \omega_0 + \alpha t$

$\theta = \omega_0 t + \frac{1}{2} \alpha t^2$

$\omega^2 = \omega_0^2 + 2 \alpha \theta$