Measuring
Rotational Motion
Chapter 7 Section 1
What is Rotational Motion?
Rotational Motion – The motion of a body
that spins about an axis.
 Examples:

 Ferris
Wheel
 Bicycle Wheel
 Merry-go-round
 Etc…
Circular Motion
A point that rotates about a single axis
undergoes circular motion around that
axis.
 Regardless of the shape of the object, any
single point on the object travels in a circle
around the axis of rotation.

Linear motion here? Think again
Its difficult to describe the motion of a point
in a circular path through linear quantities,
because the direction of the motion is not
in a straight line path.
 Therefore it is describe in terms of angle
through which the point on an object
moves.

Rotational Motion

When rotational motion is described using
angles, all points on a ridged rotating
object, except the points on the axis move
through the same angle during any time
interval.
r
θ
r = radius
 θ = angle
 s = arc length

s
Reference Line
Radians
Radian – An angle whose arc length is
equal to its radius, which is approximately
equal to 57.3º
 The radian is a pure number with no
dimensions.
 In almost all of the equations in chapter 7
and chapter 8 will use radians instead of
degrees.

Radian Chart
Radian Equations

Any angle θ, measured in radians, is defined as
the following:
s

r

To convert degrees to radians, or vise versa,
use the following equation:
  
 (rad )  
  (deg)
 180 
Angular Displacement
Angular Displacement – The angle
through which a point, line, or body is
rotated in a specific direction and about a
specified axis.
 In simple terms, angular displacement
describes how much an object has
rotated.

 It
is not a distance!
Angular Displacement Equation
s
 
r

Δθ = angular displacement (in radians)
 Units
for angular displacement: rad

Δs = change in arc length
r = distance from axis

When Δθ rotates:

 Clockwise – Negative
 Counter-clockwise – Positive
Recall
Recall that Δ is a capital Greek letter,
“delta”
 It means, “Change in”
 It is always the final minus the initial

 Example:
 Δθ
= θf-θi
Example Problem

Earth has an equatorial radius of
approximately 6380km and rotates 360º
in 24 hours.
What is the angular displacement (in
degrees) of a person standing on the
equator for 1.0 hour?
2. Convert this angular displacement to radians
3. What is the arc length traveled by this
person?
1.
Example Problem Answer
1.
2.
3.
15º
0.26 rad
Approximately 1700km
Example Problem

Earth has an average distance from the
sun of approximately 1.5x10^8 km. For
its orbital motion around the sun, find the
following:
Average daily angular displacement
2. Average daily distance traveled
1.
Example Problem Answer
1.
2.
1.72x10^-2 rad
2.58x10^6 km
Angular Speed
Angular Speed – The rate at which a body
rotates about an axis, usually expressed in
radians per second. (rad/s)
 Recall that linear speed describes the
distance traveled in a specific time, here it
is the angular displacement in a specific
time.

 It
describes how quickly the rotation occurs.
Angular Speed Equation
 avg



t
ω = Average angular speed
 Lower
case Greek letter called, “omega”
 Units: radians per second (rad/s)
Δθ = Angular displacement
 Δt = Time interval

Revolutions

Occasionally angular speeds are given in
revolutions per unit time.
1

rev = 2πrad
Examples:
 DVD’s
 Records
 Engines

2000 rpm (revolutions per minute)
Example Problem

An Indy car can complete 120 laps in 1.5
hours on a circular track. Calculate the
average angular speed of the Indy car.
Example Problem Answer

0.14 rad/s
Angular Acceleration

Angular Acceleration – The time of change
of angular speed, expressed in radians per
second per second.
Angular Acceleration Equation
 avg 

 f  i
t


t
α = average angular acceleration
 Lower
case Greek letter, “alpha”
 Units: rad/s²
Δω = Change in angular speed
 t = time interval

Example Problem

A yo-yo at rest is sent spinning at an
angular speed of 12 rev/s in 0.25 seconds.
What is the angular acceleration of the yoyo?
Example Problem Answer

48 rev/s²
 Converted

to radians
301.6 rad/s²
All Points Have Same Speed and
Acceleration

If a point on the rim of a bicycle wheel had
a greater angular speed than a point
closer to the axel, the shape of the wheel
would be changing.
 Which
can not happen in normal everyday
riding.

All points on a rotating object have the
same angular speed & acceleration.
Comparing Linear and Angular
Quantities
Angular Substitutes for Linear Quantities
Angular
Linear
Displacement
θ (units: rad)
x (units: m)
Speed
ω (units: rad/s)
v (units: m/s)
Acceleration
α (units: rad/s²) a (units: m/s²)
Kinematic Equations Can be Used
for Circular Motion
The same equations used for linear motion
with constant acceleration, can be used for
circular motion with constant angular
acceleration.
 Change the variables, by using the table in
the previous slide and you have the new
kinematic equations.

Kinematic Equations for Constant
Angular Acceleration
Angular
 f  i   t
  i t  t
1
2
Linear
v f  vi  at
2
 f  i  2
2
2
x  vi t  at
1
2
2
v f  vi  2ax
2
2
Example Problem

A barrel is given a downhill rolling start of
1.5rad/s at the top of a hill. Assume a
constant angular acceleration of
2.9rad/s²
If it takes 11.5s to get to the bottom of the
hill, what is the final angular speed of the
barrel?
2. What angular displacement does the barrel
experience during the 11.5s ride?
1.
Example Problem Answers
35 rad/s
2. 210 rad
1.