Circular Motion

If a particle moves in a circular path around a fixed point, the motion is called circular motion. If the angular velocity remains constant during this circular motion, it is called uniform circular motion.

Angular Displacement

The angle formed at the center of the circle between the initial and final positions of a rotating particle is called angular displacement.
Unit: Radian (rad)
Nature: It is a dimensionless physical quantity. It has a unit but no dimension. It is an axial vector.

Relation between Angular and Linear Displacement

\[ S = r\theta \] Linear Displacement = Angular Displacement × Radius of the circular path

Angular Velocity

The rate of change of angular displacement with respect to time is called angular velocity.
\[ \omega = \frac{\theta}{t}, \quad \omega = \frac{d\theta}{dt} \] Unit: rad/s
Dimension: [T⁻¹]
Conversions:
1 rph = \( \frac{2\pi}{3600} \) rad/s
1 rpm = \( \frac{2\pi}{60} \) rad/s
1 rps = \( 2\pi \) rad/s

Relation between Linear and Angular Velocity

\[ V = \omega \times r \] \[ \vec{V} = \vec{\omega} \times \vec{r} \] Linear Velocity = Angular Velocity × Radius

Angular Acceleration

The rate of change of angular velocity with respect to time is called angular acceleration.
\[ \alpha = \frac{\omega_2 - \omega_1}{t}, \quad \alpha = \frac{d\omega}{dt} \] Unit: rad/s²
Dimension: [T⁻²]

Relation between Linear and Angular Acceleration

\[ \vec{a} = \vec{\alpha} \times r \] Linear Acceleration = Angular Acceleration × Radius

Linear vs Angular Motion

Linear Motion Angular Motion
\( v = u + at \) \( \omega_2 = \omega_1 + \alpha t \)
\( S = ut + \frac{1}{2}at^2 \) \( \theta = \omega_1 t + \frac{1}{2} \alpha t^2 \)
\( v^2 = u^2 + 2as \) \( \omega_2^2 = \omega_1^2 + 2 \alpha \theta \)

Centripetal Acceleration and Force

Acceleration: \[ a = \frac{v^2}{r} \]
Centripetal Force: \[ F = \frac{mv^2}{r} \]

Centrifugal Force

\[ F = \frac{mv^2}{r} \]
Centrifugal force is a fictitious force — it does not exist in the inertial frame of reference. It is not an action-reaction force but is experienced only in a rotating (non-inertial) frame.

Motion of a Cyclist on Circular Path

If the cyclist makes an angle \(\theta\) with the vertical while moving with speed V in a circular path of radius r:
\[ \tan \theta = \frac{v^2}{rg} \] Maximum velocity without slipping:
\[ V_{max} = \sqrt{ \mu gr } \] Where \(\mu\) = coefficient of friction between the tire and road

Banking of Roads

If the banking angle is \(\theta\), road width is x, radius of curvature is r, and vehicle speed is v:
\[ \tan \theta = \frac{h}{x} \]
\[ \frac{v^2}{rg} = \frac{h}{x} \Rightarrow v = \sqrt{ \frac{hrg}{x} } \]

Condition for No Slipping

\[ v_m = \left( rg \cdot \frac{\mu - \tan \theta}{1 - \mu \tan \theta} \right)^{1/2} \]