Chapter 10.1 – The radian
• In this chapter we will study the rotational motion of rigid bodies about
a fixed axis. To describe this type of motion we will introduce the
following new concepts:
• Angular displacement (symbol: Δθ)
• Avg. and instantaneous angular velocity (symbol: ω )
• Avg. and instantaneous angular acceleration (symbol: α )
• Rotational inertia, also known as moment of inertia (symbol I )
• The kinetic energy of rotation as well as the rotational inertia;
• Torque (symbol τ )
• We will have to solve problems related to the above mentioned
concepts.
The radian
• We first need to get a good understanding of what a radian is and how we
can convert between radians and degrees, before we get to the content of this
chapter.
• We all know that there are 360° in a circle. But why 360° and not 400° for
instance. This system is coming from the ancient Babylonian system where
they have used a base of 60. We have since switched to the decimal system
where we use a base of 10. We need a more universal way to define angles
and we are going to use the radian for that.
The radian
Visit khanacademy for a visual explanation of the radian.
https://www.khanacademy.org/math/algebra
2/x2ec2f6f830c9fb89:trig/x2ec2f6f830c9fb89:
radians/v/introduction-to-radians
The radian
• Consider a circle with radius r.
r
The radian
• Take a length that is the same
length as the radius r.
r
r
The radian
• Give this length
to Superman.
The radian
• Ask him to bend
it to fit on the
circle.
The radian
• Now place it on
the circle.
The radian
r
• You will agree
that the length of
the blue line is
the same as that
of the red line.
r
The radian
• Continue with this
until you fill the
whole circle. You
will notice that
you still need a
piece of the red
line to complete
the circle.
The radian
• Take a piece of
the red line that
will fill the last
part.
The radian
The radian
The radian
How many radii fit around a circle?
2
3
1
?
4
5
6
6
1
4
The radian
2
3
1
?
4
5
6
Is there a more accurate way
to calculate the number of radii?
The radian
Circumference measures the distance around a circle
2
3
1
?
4
5
6
C  2 r
C   2  r
C  6.28 r
The radian
One radian is the measure of a central angle θ that intercepts an arc “s”
equal in length to the radius of the circle.
Since the circumference of a circle is C = 2πr, it takes 2π radians to get
completely around the circle once. Therefore, it takes π radians to get
halfway around the circle.
Radian vs degrees

2

2
3
3
3 
90
60
120
4
4
45
135

5
30 6
6 150
 180
0 0
y
7 210
330 11
6
6
315
5 225
7
240
300
4 4
270
5 4
3
3
3
2
x
Conversion
Between radians (rad) and degrees:
2π radians (rad) = 360°
1 radian (rad) = 57.3°
Between radians (rad) and revolutions:
2π radians (rad) = 360° = 1 revolution (rev)
1 radian (rad) = 0.159 revolutions
Conversion
•Since 180° = π radians, it follows that:
•1° =
𝜋
180°
radians and 1 radian =
180°
𝜋
• which lead to the following conversion rules.
• To convert degrees to radians, multiply degrees by
 radians
180
180
• To convert radians to degrees, multiply radians by  radians
Conversion
11
1. 220   radians   11 radians


9
 180 
9
6  180 
2.
 154.3


7   radians 