## Laws of Motion

# Dynamics of Circular Motion

- The force required to move a particle in circular path called centripetal force \tt F = \frac{mv^{2}}{r} = mr \omega^{2}
- Centripetal force always acts towards centre.
- Centripetal force is a real force.
- Formula for centripetal acceleration \tt ac = \frac{v^{2}}{r} = r \omega^{2}
- The force which imagined away from centre in Non inertial frame is centrifugal force.
- Translation equilibrium is the resultant of all the forces acting on a body keeps it in equilibrium
- Velocity of centre of mass = 0 in translational equilibrium.
- Torque is defined as the product of force and perpendicular distance τ = F × d⊥.
- When resultant torques acting on a body in equilibrium (keeps particle at rest) particle is said to be in Rotational Equilibrium.
- Frictional force (Limiting) when car is moving in a circle F
_{L}= μ m g. - Maximum safe velocity to take turn (circular) \tt V = \sqrt{\mu g r}
- Banking angle of roads without friction tanθ = \tt \frac{v^{2}}{gr}
- Maximum velocity of banking without friction \tt v = \sqrt{g r \tan \theta}
- Angular velocity of conical pendulum \tt \omega = \sqrt{\frac{g \tan \theta}{r}}
- Velocity of conical pendulum \tt V = \sqrt{g r \tan \theta}
- Time period of conical pendulum \tt \tau = 2 \pi \sqrt{\frac{L \cos \theta}{g}}
- Net force on a simple pendulum bob \tt F = m \sqrt{g^{2} \sin^{2} \alpha + \frac{V^{4}}{L^{2}}}
- Angular velocity of ball in bowl \tt w=\sqrt{\frac{g}{R \cos \alpha}}

### View the Topic in this video From 0:21 To 12:28

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1. Bending of cyclist : Angle θ of bending form vertical position is given by

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

2. Motion of a car on a level road: The maximum velocity with which a car can take a circular path of radius *r* without slipping is given by

\nu_{max} = \sqrt{\mu_{s}rg}

3. Motion of a car on a banked circular road: The maximum permissible speed to avoid slipping,

\nu_{max} = \left[\frac{rg(\mu_{s} + \tan \theta)}{1 - \mu_{s} \tan \theta}\right]^{1/2}

4. If banked road is perfectly smooth, then \nu_{0} = \sqrt{rg \tan \theta}