Download U2_8_radians

Radians
Radian measure is an alternative to degrees and is based
upon the ratio of
arc length
radius
a
ie

r
 (radians) = a/r
 - theta
If the arc length = the radius
ie
r

 (radians) = r/r = 1
r
If we now take a semi-circle
a
Here a = ½ of circumference
= ½ of d
= r
ie

r
So  (radians) =
r /
r
=
Since we have a semi-circle the the angle must be 180.
We now get a simple connection between degrees
and radians.
 (radians) = 180
This now gives us
2 = 360
/ = 90
3/ = 270
2
2
/
3
= 60
/
4
= 45
/
6
= 30
2/
3
= 120
3/
4
= 135
5/
6
= 150
etc
NB: radians are usually expressed as fractional multiples of .
Converting
180
X 
degrees
radians
X 180

The fraction button on your calculator
can be very useful here
a b/ c
Ex1
72 =
Ex2
330
Ex3
Ex4
72/
180
330/
=
2
/9 =
23
/18 =
X  =
180
2
2
X  =
/5
11
/6
/9 X 180 =
23
2/
/18 X 180 =
9
X 180
23/
18
= 40
X 180 = 230
Example 5
Angular Velocity
In the days before CDs the most popular format for music
was “vinyls”.
Singles played at 45rpm while albums played at 331/3 rpm.
rpm =revolutions per minute !
Going back about 70 years an earlier version of vinyls
played at 78rpm.
Convert these record speeds into “radians per second”.
NB:
1 revolution = 360 = 2 radians
1 min = 60 secs
So
45rpm = 45 X 2 or 90 radians per min
= 90/60 or 3/2 radians per sec
So
331/3rpm = 331/3 X 2 or 662/3  radians per min
=
So
662/3 /60
or
10/
9
radians per sec
78rpm = 78 X 2 or 156 radians per min
=
156/
60
or
13/
5
radians per sec