Radians

In mathematics, we often express angles in radians. A radian is a way of measuring angles based on the length of an arc. Specifically, the angle θ in radians is defined as the ratio of the arc length to the radius (r) of the circle:

$θ = \frac{\text{arc length}}{\text{r}}$

For a full circle, which is 360°, the arc length is the same as the circle's circumference, given by 2πr. If we divide this by the radius (r), we find:

2π radians = 360°

This tells us that a full circle is equivalent to 2π radians.

Worked example - Converting degrees to radians

Convert 30° to radians.

Step 1: Conversion Formula

To convert from degrees into radians, multiply by $\frac{\pi}{180}$

$\text{radians = }\frac{\pi}{180^\circ} \times \text{angle in degrees}$

Step 2: Substitute the Given Value

$\text{radians = }\frac{\pi}{180} \times 30 = \frac{\pi}{6}$

Angular velocity measures rotational speed

Angular velocity (ω) measures the rate at which an object rotates, or changes its angle, over time. Its formula is:

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

Where:

$\omega$ = angular velocity (rad s^-1)

$\theta$ = angle (rad)

t = time (s)

Linking linear and angular velocities

In circular motion, linear velocity (v) and angular velocity (ω) are related as follows:

$\text{v = r }\omega$

Where:

v = linear velocity (m s^-1)

r = radius (m)

$\omega$ = angular velocity ( rad s^-1)

Worked example - Calculating angular velocity

Calculate the angular velocity of a Ferris wheel that rotates 3 times in 2 minutes.

Step 1: Formula

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

Step 2: Calculate Change in Angle (Δθ)

$\Delta\theta \text{ = 3 × 2 }\pi = 6\pi \text{ radians}$

Step 3: Convert minutes to seconds (Δt)

To convert from minutes to seconds, multiply by 60

2 minutes = 120 seconds

Step 4: Substitution and correct evaluation

$\omega=\frac{6\pi}{120}\text{ = 0.157 rad s}^{-1}$

Frequency and period definitions

The frequency of rotation and the time period of rotation are related as follows:

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

Where:

f = number of complete rotations per second (Hz)

T = time taken for one complete rotation (s)

Relating f, T and ω

For each revolution, an object rotates through an angle of 2π radians.

Hence:

• f revolutions per second equals ω radians per second
• ω = 2πf
• $\omega=\frac{2\pi}{T}$

Worked example - Relating frequency, period, and angular velocity

Determine the angular velocity of a ceiling fan completing 120 revolutions per minute.

Step 1: Convert rpm to Frequency (f)

to convert from rpm to frequency, divide by 60

120 rpm = 2 rev s^-1

Step 2: Formula

$\omega\text{ = 2 } \pi \times \text{ f}$

Step 3: Substitution and correct evaluation

$ω \text{ = 2 }\pi \times \text{ 2 = 4 }\pi \text{ rad s}^{-1}$