## Motion in a Plane

# Kinematics of Circular Motion

- The angle swept by the radius vector is called angular displacement the unit of angular displacement is radian.
- The angular velocity is the rate of change of angular displacement in unit time. The unit of angular velocity is radian per second W = dθ/dt
- The angular acceleration is the rate of change of angular velocity in unit time. The unit of angular acceleration is radian per second. ∝ = dw/dt
- The relation between Linear velocity and angular velocity is v = \overline{w} \times \overline{r} where \overline{r} is the position vector of the particle with respect to centre of circle.
- The relation between Linear acceleration and angular acceleration is \overline{a} = \overline{\alpha} \times \overline{r} where \overline{r} is the position vector of the particle with respect to centre of circle
- Tangential acceleration is the component of acceleration in the direction of velocity which is responsible for change in speed of particle (at = dv/dt) Thin component is tangential to the circle
- Radial acceleration is the component of acceleration in a direction towards centre. This component is responsible for change in direction of Velocity. a_{r} = \frac{v^{2}}{r} = rw^{2}
- The final angular velocity wf = wi + αt when α is the angular acceleration and 't' is the time and wi is the initial angular velocity.
- The final angular velocity wf
^{2}= wi^{2}+ 2 αθ where 'α' is the angular acceleration and 'θ' is the angular displacement. - The angular displacement \theta = wit + \frac{1}{2} \alpha t^{2} where 'wi' is the initial angular velocity 'α' is the angular acceleration at time 't'.

### View the Topic in this video From 20:29 To 29:33

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1. Angular displacement (Δθ) = \frac{\Delta s}{r}

2. Angular velocity (\omega) = \frac{\Delta \theta}{\Delta t}

3. Angular acceleration (\alpha) = \frac{d \omega}{dt} = \frac{d^{2}\theta}{dt^{2}}

4. Centripetal acceleration \alpha = \frac{v^{2}}{r} = r\omega^{2}

5. Centripetal force F = \frac{mv^{2}}{r} = mr\omega^{2}

6. Kinematical Equations in Circular Motion

(i) ω = ω_{0} + α*t *(ii) \theta = \omega_{0}t + \frac{1}{2}\alpha t^{2} (iii) \omega^{2} = \omega_{0}^{2} + 2\alpha \theta

7. The coefficient of friction (μ_{s}) between the road and tyres should be, \mu_{s} \geq \frac{v^{2}}{rg} \ {\tt or} \ v \leq \sqrt{\mu_{s} rg}

8. If centripetal force is obtained only by the banking of roads, then the speed (v) of the vehicle for a safe turn v = \sqrt{rg \tan \theta}

9. When centripetal force is obtained from friction force as well as banking of roads, then the maximum safe value of speed of vehicle

v_{max} = \sqrt{\frac{rg (\tan \theta + \mu)}{(1 - \mu \tan \theta)}}

10. If a cyclist inclined at an angle θ, then \tan \theta = \frac{v^{2}}{rg}

11. When a vehicle is moving over a convex bridge, then at the maximum height, reaction (N_{1}) is

N_{1} = mg - \frac{mv^{2}}{r}

12. When a vehicle is moving over a concave bridge, then at the lowest point, reaction (N_{2}) is

N_{2} = mg + \frac{mv^{2}}{r}

13. Both acceleration acts perpendicular to each other. Resultant acceleration

a = \sqrt{a_{R}^{2} + a_{T}^{2}} = \sqrt{\left[\frac{v^{2}}{r}\right]^{2} + (r\alpha)^{2}}

and \tan \phi = \frac{a_{T}}{a_{R}} = \frac{r^{2}\alpha}{v^{2}}

14. Time period of conical pendulum, T = 2\pi\sqrt{\frac{l \cos \theta}{g}}