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.

Go to
Angular velocity game

math expression

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:

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

Answer

s = rθ

= 4 × 5.1

= 20.4 cm

Area of a Sector

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

math expression

If r is in m, the area will be measured in m². If r is in cm, area will be in cm².

Example:

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

Answer

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:

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?

Answer

math expression

Now v = r ω

The units are a mix of cm and km. Let's present everything in meters.

We need to convert v to m/s first.

25 km/h = 25000 m/h = 25000/3600 m/s = 6.94444 m/s

Also, we have r = (90 cm)/2 = 45 cm = 0.45 m

So ω = v/r = 6.94444/0.45 = 15.43 rad/s

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

Your points go down if:

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.

Loading Flash movie...

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?

Answer

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?

Answer

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

Answer

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?

Answer

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

Answer

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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