Circular Motion
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: $\omega_0$
$$\omega = \omega_0 + \alpha t$$
$$\theta = \omega_0 t + \frac{1}{2} \alpha t^2$$
$$\omega^2 = \omega_0^2 + 2 \alpha \theta$$
