Circular Motion

1. Introduction

A body is said to be in circular motion when it moves along a circular path. Many important motions in nature and technology are circular or nearly circular: a stone whirled on a string, a car turning a bend, a satellite orbiting the Earth, and the electrons pictured as circling a nucleus. When a body covers equal arcs of the circle in equal intervals of time it is in uniform circular motion: the speed (magnitude of velocity) is constant, but the direction of the velocity changes continuously, so the motion is accelerated.

To describe circular motion we use angular quantities (angular displacement, angular velocity, angular acceleration) alongside the familiar linear quantities, and we relate the two sets. We then study the acceleration that is always directed towards the centre (centripetal acceleration), the force that produces it (centripetal force), and several standard applications: motion in a vertical circle, a vehicle on a banked road, and the conical pendulum.

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2. Angular Displacement, Angular Velocity and Angular Acceleration

2.1 Angular displacement

When a body moves along a circular path, the angle swept out by the radius line joining the body to the centre is called the angular displacement. It is denoted by $\theta$.

If a body moves along an arc of length $s$ on a circle of radius $r$, the angular displacement is defined as:

$$\theta = \frac{s}{r}$$

Its SI unit is the radian (rad), and it is a dimensionless quantity (a ratio of two lengths). One complete revolution corresponds to an angular displacement of $2\pi$ radians.

2.2 Angular velocity

The rate of change of angular displacement with time is called the angular velocity. It is denoted by $\omega$.

The average angular velocity over a time interval $\Delta t$ is $\omega_{av} = \dfrac{\Delta\theta}{\Delta t}$, and the instantaneous angular velocity is:

$$\omega = \frac{d\theta}{dt}$$

Its SI unit is radian per second ($\text{rad s}^{-1}$) and its dimensional formula is $[\mathrm{T^{-1}}]$. Angular velocity is a vector directed along the axis of rotation (given by the right-hand rule).

2.3 Angular acceleration

The rate of change of angular velocity with time is called the angular acceleration. It is denoted by $\alpha$.

$$ \begin{aligned} \alpha &= \frac{d\omega}{dt} \ &= \frac{d^2\theta}{dt^2} \end{aligned} $$

Its SI unit is radian per second squared ($\text{rad s}^{-2}$) and its dimensional formula is $[\mathrm{T^{-2}}]$. In uniform circular motion $\omega$ is constant, so $\alpha = 0$.

2.4 Frequency and time period

• Time period ($T$): the time taken by the body to complete one full revolution.
• Frequency ($f$): the number of complete revolutions made by the body in one second. Its SI unit is the hertz (Hz).

Frequency and time period are reciprocals of each other:

$$f = \frac{1}{T}$$

Since one complete revolution is an angular displacement of $2\pi$ radians covered in time $T$, the angular velocity is:

$$ \begin{aligned} \omega &= \frac{2\pi}{T} \ &= 2\pi f \end{aligned} $$

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3. Relation Between Angular and Linear Quantities

3.1 Linear velocity and angular velocity

Consider a body moving along a circular path of radius $r$. Suppose it starts at point $A$ and after time $t$ reaches point $B$, having swept an angular displacement $\theta$ while covering an arc (linear displacement) $s$.

From the definition of angular displacement:

$$s = r\theta \qquad \text{(i)}$$

Differentiating equation (i) with respect to time, and noting that the radius $r$ is constant:

$$\frac{ds}{dt} = r,\frac{d\theta}{dt} \qquad \text{(ii)}$$

Here $\dfrac{ds}{dt} = v$ is the linear velocity and $\dfrac{d\theta}{dt} = \omega$ is the angular velocity. Therefore:

$$\boxed{v = r\omega} \qquad \text{(iii)}$$

This is the relation between linear velocity and angular velocity. In vector form the linear velocity is $\vec{v} = \vec{\omega}\times\vec{r}$.