Physics > Circular Motion > 1.0 Introduction
Circular Motion
1.0 Introduction
1.1 Angular Variables
1.2 Kinematic equation for circular motion
1.3 Relation between angular and linear variables
1.4 Unit vectors along the radius and tangent
1.5 Velocity and acceleration of particle in circular motion
2.0 Dynamics of circular motion
3.0 Motion in a vertical circle
4.0 Rigid body rotating in a vertical circle
5.0 Circular turning of roads
6.0 Conical Pendulum
7.0 Death well
8.0 Rotor
9.0 Bending of a cyclist or motorcyclist while taking turn
10.0 Centrifugal force
1.1 Angular Variables
1.2 Kinematic equation for circular motion
1.3 Relation between angular and linear variables
1.4 Unit vectors along the radius and tangent
1.5 Velocity and acceleration of particle in circular motion
Let us assume a particle moving in circular path of radius $r$ with center $O$.
The particle is at point $P$, whose position at any time $t$ is described by the angle $\theta $. The angle $\theta $ is called the angular position of particle at point $P$.
As the particle moves on the circle it’s angular position $\theta $ changes.
Let the particle rotates an angle $\Delta \theta $ in time $\Delta t$ about its center $O$.
The rate of change of angular position is known as the angular velocity $\left( {\overrightarrow \omega } \right)$.
Thus,
$$\begin{equation} \begin{aligned} \overrightarrow \omega = \mathop {\lim }\limits_{\Delta t \to 0} \frac{{\Delta \overrightarrow \theta }}{{\Delta t}} \\ \overrightarrow \omega = \frac{{d\overrightarrow \theta }}{{dt}} \\\end{aligned} \end{equation} $$
The rate of change of angular velocity is known as the angular acceleration $\left( {\overrightarrow \alpha } \right)$. Thus, $$\begin{equation} \begin{aligned} \overrightarrow \alpha = \mathop {\lim }\limits_{\Delta t \to 0} \frac{{\Delta \overrightarrow \omega }}{{\Delta t}} \\ \overrightarrow \alpha = \frac{{d\overrightarrow \omega }}{{dt}} \\\end{aligned} \end{equation} $$ or $$\begin{equation} \begin{aligned} \overrightarrow \alpha = \frac{d}{{dt}}\frac{{d\overrightarrow \theta }}{{dt}} \\ \overrightarrow \alpha = \frac{{{d^2}\overrightarrow \theta }}{{d{t^2}}} \\\end{aligned} \end{equation} $$
