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# Main concepts of circular motion

## Uniform and non-uniform circular motion

Circular motion is the movement of an object on a circular path. This means that the movement of the object traces the arc of a circle (as opposed to following an arbitrary curved path).

Circular motion can be classified into uniform and non-uniform circular motion:

• An object undergoing uniform circular motion moves with a constant speed (only the direction changes).

• An object undergoing non-uniform circular motion moves with varying speed.

Unless otherwise mentioned, it is assumed (by convention) that objects undergoing circular motion do so uniformly (i.e. with constant speed).

A satellite orbits the earth in circular motion.

## Newton's laws and circular motion

Newton's first law states that an object will continue to move in a straight line with the same velocity if it is not affected by a force. Therefore, an object undergoing circular motion must have a force acting on it that changes its direction.

Newton's second law states that an object under the influence of a force will accelerate in the direction of the force.

This means that the force on the object cannot have components that are in the direction of the velocity. The force must therefore always be perpendicular to the velocity of the object (i.e. pointing to the centre of circular motion).

Acceleration in uniform circular motion is therefore perpendicular to the trajectory. In non-uniform circular motion, a non-zero component remains in the direction of motion.

In the case of non-uniform circular motion, the acceleration is not perpendicular to the motion.

## Angular displacement

An object moving in circular motion traces the arc of a circle. The angle between the radii at the beginning and the end of the arc (the angle subtended by the arc) is the angular displacement $(\theta)$ of the object.

This is given by $\vecphy{\theta}=\frac{\vecphy{s}}{r}$, where $s$ is the arc length and $r$ is the radius of the arc.

Angular displacement can be thought of as a vector, but for simplicity it is usually considered to be a scalar.

The SI unit of angular displacement is the radian $(\text{rad})$. One radian is $180/\pi$ degrees.

The angular displacement can exceed $2\pi\text{ rad}$, which means that more than one revolution has been completed. The angular displacement can therefore be thought of as an angular "distance."

An object moving in uniform circular motion with angular displacement $\vecphy{\theta}$, angular velocity $\vecphy{\omega}$, velocity $\vecphy{v}$ and radius $r$

## Angular velocity

The angular velocity $(\omega)$ is the rate of change of angular displacement (i.e. the change in angle per unit time). It is a vector quantity.

The angular velocity gives the rate at which an arc is traced by an object undergoing circular motion.

This is given by the equation below, where $t$ is the time taken for the object to achieve an angular displacement of $\theta$.$$\vecphy{\omega}=\frac{\vecphy{\theta}}{t}$$

The SI unit of angular velocity is radian per second $(\text{rad s}^{-1})$.

The angular velocity is related to the tangential linear velocity $\vecphy{v}$ (normally just referred to as the "velocity") by: $$\vecphy{v}=r\vecphy{\omega}$$ Tangential velocity has the same units as linear velocity (i.e. $\text{m/s}$).

An object moving in uniform circular motion with angular displacement $\vecphy{\theta}$, angular velocity $\vecphy{\omega}$, velocity $\vecphy{v}$ and radius $r$

## Period and frequency of circular motion

The period $(T)$ of circular motion is the time taken for an object undergoing circular motion to complete one full revolution of the circle.

The frequency $(f)$ of circular motion is the number of revolutions completed by the object per second. It is related to the period by:$$f=\frac{1}{T}$$The SI unit of frequency is Hertz $(\text{Hz})$. The unit of Hertz is equivalent to one inverse second (i.e. $1\text{ Hz}=1\text{ s}^{-1}$).

The angular speed $\omega$ can be obtained from the period and frequency by:$$\omega=\frac{2\pi}{T}=2\pi f$$This formula is derived from the formula $\omega=\theta/t$ by taking $\theta=2\pi$ (one revolution) as the angle and $t=T$ (one period) as the time. This formula only holds for objects undergoing uniform circular motion.

An object moving in uniform circular motion with angular displacement $\vecphy{\theta}$, angular velocity $\vecphy{\omega}$, velocity $\vecphy{v}$ and radius $r$