# 8. Applications of Radian Measure

by M. Bourne

### Linear velocity applet

Don't miss the interactive angular/linear velocity animation on this page.

Go to

Angular velocity applet

In this section, we see some of the common applications of radian measure, including arc length, area of a sector of a circle, and angular velocity.

Go back to the section on Radians if you are not sure what is going on.

## Arc Length

The length, *s*, of an arc of a circle radius *r*
subtended by *θ* (in **radians**) is given by:

s=rθ

If *r* is in meters, *s* will also be in meters. Likewise, if *r* is in cm, *s* will also be in cm.

### Example 1

Find the length of the arc of a circle with radius `4\ "cm"` and central angle `5.1` radians.

## Area of a Sector

Area, *A*, of a sector of a circle.

The area of a sector with central angle *θ* (in radians) is
given by:

`"Area"=(theta\ r^2)/2`

If *r* is measured in `"cm"`, the area will be in `"cm"`^{2}. If *r* is in `"m"`, the area will be in `"m"`^{2}.

### Example 2

Find the area of the sector with radius `7\ "cm"` and central angle `2.5` radians.

## Angular Velocity

The time rate of change of angle *θ* by a rotating body is the
**angular velocity**, written *ω* (omega). It is measured in
radians/second.

If *v* is the **linear velocity** (in m/s) and
*r* is the radius of the circle (in m), then

v=rω

**Note: **If *r* is in `"cm"`, *v* will be in `"cm/s"`.

### Example 3

A bicycle with tyres `90\ "cm"` in diameter is travelling at `25` km/h. What is the angular velocity of the tyre in radians per second?

## Interactive linear velocity applet

### Background

A car is going around a circular track of radius 0.5 km. The speedometer in the car shows the (magnitude) of the linear velocity.

At first, the car goes around the track once in just over 6 minutes. It's **angular velocity** is 1 rad/min or one complete revolution in 2π = 6.28 min.

The **distance** travelled in this time is the circumference of the circle, C = 2πr = 2π(0.5) = 3.14 km. So the car is travelling at `(3.14" km")/(6.28 min) = 0.5" km/min" = 30" km/h"`.

The **linear velocity** showing on the speedo is 30 km/h.

### Things to do

In this applet, you can:

- Vary the
**radius**of the track - Vary the
**angular velocity**of the car

Observe the change in linear speed as you do so.

Of course, angle measures are in radians in this applet.

*r = *0 km *ω = *0 rad/min

*v = rω* *=* 0 × 0
*=* 0 km/min *=* 0 km/h

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### Exercises:

1. A section of side walk is a circular sector of radius `1.25\ "m"` and central angle `50.6°`. What is the area of this section of sidewalk?

2. A cam is in the shape of a circular sector with radius `1.875\ "cm"` and central angle `165.58°`. What is the perimeter of the cam?

3. The roller on a computer printer makes `2200` rev/min. What is its angular velocity?

4. The propeller on a motorboat is rotating at `130` rad/s. What is the linear velocity of a point on the tip of a blade if the blade is `22.5` cm long?

5. The sweep second hand of a watch is `15.0` mm long. What is the linear velocity of the tip?

## Pulley Problems

You can investigate the linear velocity of a belt moving around two pulleys in this interactive example.

Go to Pulleys simulation.

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