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
1.3 Relation between angular and linear 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
In the figure, a circular path of radius $r$ is shown. Arc $AB$ of length $\Delta s$ subtends an angle of $\Delta \theta $ at the center.
We have following three relations between angular and linear variables,
- $\Delta s = r\Delta \theta \quad {\text{or}}\quad s = r\theta $
- $\Delta v = r\Delta \omega \quad {\text{or}}\quad v = r\omega $
- $\Delta a = r\Delta \alpha \quad {\text{or}}\quad a = r\alpha $
Let us derive the above relations for better understanding.
1. $\Delta s = r\Delta \theta \quad {\text{or}}\quad s = r\theta $
The length circumference of the above circular path is $2\pi r$ ands subtends an angle of $2\pi $ at the center.
Therefore by simple unitary method we can find the relation between $\Delta s$ and $\Delta \theta $.
$2\pi $ radians subtends an arc of length $2\pi r$
1 radian subtends an arc of length $\left( {\frac{{2\pi r}}{{2\pi }}} \right)$
$\Delta \theta $ radians subtends an arc of length, $\Delta s = \frac{{2\pi r}}{{2\pi }} \times \Delta \theta $
So, $$\Delta s = r\Delta \theta \quad {\text{or}}\quad s = r\theta $$
2. $\Delta v = r\Delta \omega \quad {\text{or}}\quad v = r\omega $
We know that, $$s = r\theta $$
Differentiating the above equation with respect to time $t$,
$$\begin{equation} \begin{aligned} \frac{{ds}}{{dt}} = r\frac{{d\theta }}{{dt}} \\ v = r\omega \\\end{aligned} \end{equation} $$
3. $\Delta a = r\Delta \alpha \quad {\text{or}}\quad a = r\alpha $
We know that, $$v = r\omega $$
Differentiating the above equation with respect to time $t$,
$$\begin{equation} \begin{aligned} \frac{{dv}}{{dt}} = r\frac{{d\omega }}{{dt}} \\ a = r\alpha \\\end{aligned} \end{equation} $$
Note: In case of circular motion, all the above relations hold good.
