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# 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.

We need to find length s.

s = r θ

= 4 × 5.1

= 20.4 cm

## 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.

This is the area we need to find:

Area, a sector of a circle, radius 7 and central angle 2.5 rad.

"Area"=(theta\ r^2)/2=(2.5xx7^2)/2=61.25\ "cm"^2

## 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?

The arrows represent the linear speed of 25 km/h

We learned that linear velocity for a wheel rotating at ω rad/s is given by:

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"

## 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

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?

Circular sector, radius 1.25 m, central angle 56°.

First we must convert 50.6° to radians:

50.6° = 50.6 × π/180 = 0.8831\ "radians"

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

=(0.8831xx1.25^2)/2

=0.690\ "m"^2

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?

Circular sector, radius 1.875 cm, central angle 165.58°.

The length of the arc is given by s = rθ.

First we must convert 165.58° into radians:

165.58° = 165.58 × π/180 = 2.8899 radians.

So arc length is: s = 1.875 × 2.8899 = 5.419 cm.

So the perimeter of the cam is:

2 × 1.875 + 5.419 = 9.169 cm.

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

Angular velocity is:

ω = 2200 r/min × (2π) / 60 = 230.4\ "rad/s".

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?

Linear velocity = v = ωr

In this example, ω = 130 rad/s and r = 0.225 m

So the linear velocity is:

v = 130 × 0.225 = 29.3 ms-1.

Note 1: ms-1 is an equivalent way of writing m/s. This comes from the index laws where the rule is s^-1= 1/s.

Note 2: It is common in physics to write velocity using ms-1 and the units for acceleration as ms-2.

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

The second hand rotates 2pi every minute, so per second we have:

omega=(2pi)/60=pi/30"rad""/"s

and r= 0.015\ "m".

So

v=omega r

=(pi/30)(0.015)

 = 0.00157\ "m/s".

## 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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