The document discusses circular motion, detailing how it can be measured in linear and angular units, including properties like angular displacement and velocity. It explains the equations for angular motion and their relationship to linear equations, as well as the concepts of frequency and period in rotation. Additionally, it introduces radians and their connection to revolutions, providing formulas for calculating related quantities.

1. Sautter 2015

2. Circular Motion
• Circular motion (rotation) can be measured using
linear units or angular units. Angular units refer to
revolutions, degrees or radians.
• The properties of circular motion include
displacement, velocity and acceleration. When
applied to rotation the values become angular
displacement, angular velocity or angular
acceleration. Additionally, angular motion can be
measured using frequencies and periods or rotation.
• The Greek letters theta (), omega () and alpha ()
are used to represent angular displacement, angular
velocity and angular acceleration.
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3. 


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4. Circular Motion
• The equations which describe angular motion are
similar to those describing linear motion. Rotational
equations can therefore be derived from linear
equations by analogy (direct comparison).
• Recall the following linear motion equations:
• VAVERAGE = s/ t = (V2 + V1) / 2
• Si= V0 t + ½ at2
• Vi = VO + at
• Si = ½ (Vi 2 – Vo
2) /a
• Each of these can be converted to a rotational
motion equation by substituting the rotational
quantity in for the appropriate linear quantity.
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5. VAVERAGE = s/ t = (V2 + V1) / 2
AVERAGE =   /  t = (2 + 1) / 2
Si= V0 t + ½ at2
 = o t + ½ t2
Vi = VO + at
i = o + t
Si = ½ (Vi 2 – Vo
2) /a
I = ½ (i
2 - o
2) /  5

6. The linear velocity of a body can be related
to it angular velocity by the equation:
V =  R
Where V = linear velocity in m/s , ft/s, etc
= angular velocity in radians/s
R = the radius of the object in meters, feet, etc.
The linear acceleration of a body can be related
to it angular acceleration by the equation:
a =  R
Where a = linear acceleration in m/s2 , ft/s2, etc
 = angular acceleration in radians/s2
R = the radius of the object in meters, feet, etc.
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7. Frequency & Period
• Two other important quantities relating to circular motion are
frequency and period.
• Frequency refers to how often (frequently) an object rotates. If a
body rotates 10 complete revolutions in 2 seconds the frequency
of rotation is 10/2 or 5 revolutions per second. The unit “hertz”
is use to represent cycles or rotations per second. Therefore, 5
revolutions per second is 5 hertz abbreviated as 5 Hz.
• Period is the time for one complete cycle or revolution. If an
object rotates at 5 revolutions per second (5 Hz) the each
revolution takes 1/5 second or 0.20 seconds and the period then
is 0.20 seconds.
• The symbol for frequency is f. The symbol for period is T.
Frequency and period are related to each other by the equation:
f = 1/ T and T = 1 / f . 7

8. Revolutions & Radians
• Radians are defined as arc length divided by radius length. In a
complete circle the circumference is the arc length and is
calculated by the equation C = 2  R where R is the radius.
Dividing the arc (the circumference) by the radius, we get 2 
R / R gives 2  . The number of radians in one complete circle
then is 2  .
• 1 revolution = 360 degrees = 2  radians
• Linear distance can be calculated by multiplying the angular
displacement times the radius.
• s =  R
• Frequency is the number of revolutions per second and since each
revolution is 2  radians, radians per second can be calculated by
2  x frequency. Angular velocity is measured in radians per
second therefore, angular velocity can be calculated as 2  f.
•  = 2  f and since f = 1 / T so  = 2  / T 8

9. AVERAGE =  /  t = (2 + 1) / 2
 = o t + ½ t2
i = o + t
i = ½ (i
2 - o
2) / 
s =  R
Vlinear =  R
alinear =  R
f = 1/ T
T = 1 / f
1 revolution = 360 degrees = 2  radians
 = 2  f
 = 2  / T 9

10. Problems in Rotational Motion
A wheel 80 cm in diameter turns at 120 rpm (revolution
per second) (a) What is its angular velocity (b) What is its
linear velocity ?
• (a)  = 2  f ,  = 2  2 = 4  radians / sec
• (b) V =  R, V = 4  (80) = 320  cm / sec
80 cm
120 rpm
Time units in minutes must
be converted to seconds
(MKS)
120 rpm / 60 = 2 rps
rps (revolutions per second)
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