Laws of Motion - Dynamics of Uniform Circular Motion

Centripetal Force: For a body of mass $m$ moving in a circle of radius $r$ with constant speed $v$, the net force acting towards the center is called centripetal force, given by $F_c = \frac{mv^2}{r}$.

Centripetal Acceleration: The acceleration directed towards the center of the circular path is $a_c = \frac{v^2}{r} = \omega^2 r$, where $\omega$ is the angular velocity.

Motion on a Level Road: When a car takes a turn on a horizontal road, the necessary centripetal force is provided by the static friction between the tires and the road ($f_s \le \mu_s N$).

Maximum Velocity on a Level Road: To avoid skidding, the maximum speed is $v_{max} = \sqrt{\mu_s r g}$, where $\mu_s$ is the coefficient of static friction.

Banking of Roads: To reduce dependence on friction and prevent skidding at high speeds, the outer edge of the road is raised above the inner edge. This is called banking, and the angle it makes with the horizontal is the banking angle $\theta$.

Optimum Speed on a Banked Road: The speed at which no frictional force is required is $v_0 = \sqrt{rg \tan \theta}$.

Maximum Safe Speed on a Banked Road: Considering friction $\mu_s$, the maximum speed is $v_{max} = \sqrt{rg \left( \frac{\mu_s + \tan \theta}{1 - \mu_s \tan \theta} \right)}$.

Bending of a Cyclist: A cyclist leans inwards while taking a turn to provide the necessary centripetal force through the horizontal component of the normal reaction.