# Angles in Radians and Angular Speed versus Linear Speed

## Angles in Radians and Angular Speed versus Linear Speed

#### Angles in radians

We usually measure angles in degrees.

360° = 1 rotation

But it's not the most convenient way to measure angles in circular motion.

Here's an alternative: Radians. The radius of a circle and its circumference are related by the equation...

Circumference = 2πr

So the factor that allows you to convert from circumference (distance travelled around the arc of the circle) to radius is 2π.

So in this way, 2π describes a whole circle, just as 360° describes a whole circle.

360° ≡ 2π radians and

180° ≡ π radians

In fact, as long as you use angles in radians you can write this general equation:

s = rθ

Where:

s = arc length covered

r = radius of the circle

θ = angle in radians

Example:

Convert the following angles from degrees into radians.

90°, 135°, 330°

Answer:

180° ≡ π radians.

So multiply any angle in degrees byto find the same angle in radians.

Example:

What angle in degrees has a car travelled around a circular track if the track has a radius of 100 m and the distance covered by the car is 470 m?

Answer:

Convert to degrees:

Note: We had to turn the conversion factor upside down to convert from radians to degrees.

Question:

#### Angular speed

In linear or straight-line motion, we measure speed by looking at how much distance is covered each second. You can do that in circular motion too, but it's often better to use angular speed, ω.

Angular speed measures the angle of a complete circle (measured in radians) covered per second.

For instance,

Where:

θ = angle of rotation in radians

t = time taken in seconds.

If you consider that the time taken for a complete rotation is the period, T, then

because 2π is the angle covered (in radians) when you do a complete circle.

Remembering thatyou can also write this as

ω = 2πf

Example:

An old record player spins records at 45rpm (revolutions per minute). For a point on the circumference (radius = 10cm) calculate the angular speed in rad s-1.

Answer:

45rpm = 45/60 = 0.75 revolutions per second = f

Angular speed = ω = 2πf x 0.75 = 4.7 rad s-1

Question:

The wheel of a car rotates at 10 revolutions per second as the car travels along. The radius of the rubber on the tyre is 20cm.

#### The relationship between angular speed and linear speed

If you are going round in a circle of radius, r, and you are travelling at a linear speed, v ms-1:

The distance covered in 1 rotation = 2πr

The time for one rotation = T, the period.

These equations allow you to relate angular and linear speed.

Question:

A shot put is swung round at 1 revolution per second. The athlete's arm is 60cm long.