# 8. Applications of Radian Measure

by M. Bourne

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

### Flash Interactive

Don't miss the interactive Flash game in this section.

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 in "m", the area will be measured in "m"2. If r is in "cm", area will be in "cm"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?

## Flash Game

The man runs at a constant velocity and a ball is revolving overhead.

You increase points if:

• the linear velocity of the ball is more than 6 m/sec - add 1 point
• the ball is close to the man (within one body length) - add 10 points

• the ball goes outside the borders - minus 1 point
• the ball hurts the man - minus 20 points

You can use

• the first slider or the right and left arrows on your keyboard to change the radius and
• the second slider or the up and down arrows to change the angular velocity.

### 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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