Mathematics of circular motion (AQA A-Level Physics): Revision Notes

Mathematics of circular motion

Introduction to circular motion

When studying rotational dynamics, we begin by examining how a single particle moves in a circular path. This forms the foundation for understanding more complex rotating systems. A particle moving in a circle has both angular and linear characteristics that we need to describe mathematically.

A particle P moving in a circular path

Angular velocity and angular displacement

Defining angular velocity

When a particle moves along a circular path, its position can be described by an angular displacement ($\theta$), which represents the angle through which the radius has rotated from a reference position. This angle must be measured in radians for the equations of rotational motion to work correctly.

Angular velocity ($\omega$) describes how quickly an object rotates. It is defined as the rate of change of angular displacement with respect to time:

$\omega = \frac{\Delta\theta}{\Delta t}$

where:

• $\omega$ is the angular velocity
• $\Delta\theta$ is the change in angular displacement (in radians)
• $\Delta t$ is the time interval

Angular velocity is measured in rad s⁻¹ (radians per second).

Understanding radians

Mathematically, this gives us the relationship:

$s = r\theta$

where $s$ is the arc length, $r$ is the radius, and $\theta$ is the angle in radians.

Since the circumference of a complete circle is $2\pi r$, there are $2\pi$ radians in a full circle. This means:

$2\pi \text{ radians} = 360°$

This conversion is essential when working between degrees and radians.

Revolutions and conversions

In rotational dynamics, particularly when dealing with engines and rotating machinery, angular velocity is often expressed in revolutions per minute (rpm) rather than radians per second. One revolution (rev) is equivalent to one complete rotation of $2\pi$ radians.

To convert from rpm to rad s⁻¹, we need to:

1. Convert revolutions to radians by multiplying by $2\pi$
2. Convert minutes to seconds by dividing by 60

Therefore:

$\text{angular velocity in rad s}^{-1} = \frac{\text{rpm} \times 2\pi}{60}$

Period of rotation

The period of rotation ($T$) is the time taken for one complete revolution. It is related to angular velocity by:

$T = \frac{2\pi}{\omega}$

Worked example: converting angular velocity

Tangential velocity

Understanding tangential velocity

A particle moving in a circular path has a linear velocity at any instant, which is directed along the tangent to the circle at that point. This is called the tangential velocity ($v$). Although the particle follows a curved path, at any given moment, its instantaneous velocity is in a straight line tangent to the circle.

The tangential velocity is directly related to the angular velocity by:

$v = r\omega$

where:

• $v$ is the tangential velocity
• $r$ is the radius of the circular path
• $\omega$ is the angular velocity

Physical interpretation

You can observe this by spinning a disc on a pencil and marking dots along a radius. The dots further from the centre travel faster and cover more distance in the same time period, even though they all complete one rotation together.

Units in the relationship

The units in the equation $v = r\omega$ must be consistent. If the radius is measured in metres (m), then:

• Angular velocity must be in rad s⁻¹
• Tangential velocity will be in m s⁻¹

If the radius is measured in centimetres (cm), then the tangential velocity will be in cm s⁻¹, provided $\omega$ remains in rad s⁻¹.

Angular acceleration

Defining angular acceleration

Angular acceleration ($\alpha$) occurs when a rotating object changes its rate of rotation—either speeding up or slowing down. It is defined as the rate of change of angular velocity:

$\alpha = \frac{\Delta\omega}{\Delta t}$

where:

• $\alpha$ is the angular acceleration
• $\Delta\omega$ is the change in angular velocity
• $\Delta t$ is the time interval

Angular acceleration is measured in rad s⁻² (radians per second squared).

Distinguishing angular acceleration from centripetal acceleration

Tangential acceleration

When angular acceleration occurs, there must be a corresponding change in tangential velocity, since $v = r\omega$ and $\omega$ is changing. This gives rise to a tangential acceleration ($a$), which acts along the direction of motion (tangent to the circle).

The relationship between tangential acceleration and angular acceleration is:

$a = r\alpha$

where:

• $a$ is the tangential acceleration
• $r$ is the radius
• $\alpha$ is the angular acceleration

As with the earlier relationships, the units must be consistent. If $r$ is in metres, then:

• Angular acceleration must be in rad s⁻²
• Tangential acceleration will be in m s⁻²

This tangential acceleration represents the rate of change of the tangential velocity—the particle is speeding up or slowing down along its circular path.