# 8. Applications of Radian Measure

by M. Bourne

### Linear 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 = r θ

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

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 =

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 = = 0 × 0 = 0 km/min = 0 km/h

Angular velocity: 1

### 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 Flash example.

Go to Pulleys simulation.

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