
Previous chapters described motion
along a straight line
› Translational (linear) motion
This chapter we will focus on rotational
motion around some fixed axis.
 Motion can be more fully described by
both translational and rotational motion

› Rolling
Rolling Motion
Rolling Motion
Two views of rolling motion: 1) Pure rotation around
the instantaneous axis or 2) rotation and translation.
Rotational Motion
Need similar concepts for objects
moving in circle (CD, merry-go-round,
etc.)
 As before:

› need a fixed reference system (line)
› use polar coordinate system
Previously, we defined a displacement in
translational motion as
Δx = x – x0

In order to define displacement for
rotation we will use an angular measure
called the radian
› Θ = 1 radian ≈ 57°

The radian can be
defined as the arc
length s along a circle
divided by the radius r

Comparing degrees and
radians
360
1 ra d 
 5 7 .3 
2

Converting from degrees
to radians

 [ ra d ] 
 [d e g re e s ]
180
Angular Displacement


Every point on the object
undergoes circular motion
about the point O
Angles generally need to be
measured in radians
s

r

Note:
1 ra d 
length of arc
radius
360
 5 7 .3 
2

 [ ra d ] 
 [ d e g re e s ]
180
 Axis
of rotation is the
center of the disk
› Fixed origin O
 Need
a fixed
reference line
› Usually, x-axis
 During
time t, the
reference line moves
through angle θ
Angular Displacement

The angular displacement is
defined as the angle the object
rotates through during some time
interval
  

f

i
Every point on the disc
undergoes the same angular
displacement in any given time
interval

Rigid Body
› Every point on the object undergoes circular
motion about the point O
› All parts of the object of the body rotate
through the same angle during the same
time
› The object is considered to be a rigid body
 This means that each part of the body is fixed
in position relative to all other parts of the body

Translation Motion
 Δx = xf – xi

Rotational Motion
 Δθ = θf - θi


The unit of angular displacement is the radian
Each point on the object undergoes the same
angular displacement
Angular Velocity
The average angular
velocity (speed), ω, of a
rotating rigid object is
the ratio of the angular
displacement to the
time interval
 Analogous to linear
motion

 f   i 
 

t f  ti
t
The instantaneous angular speed is defined
as the limit of the average speed as the time
interval approaches zero
 Units of angular speed are radians/sec

› rad/s
Speed will be positive if θ is increasing
(counterclockwise)
 Speed will be negative if θ is decreasing
(clockwise)

Angular Acceleration
What if object is initially at rest and
then begins to rotate?
 The average angular acceleration,
a, of an object is defined as the
ratio of the change in the angular
speed to the time it takes for the
object to undergo the change:

 f   i 
a 

t f  ti
t
Units are rad/s²
 Similarly, instant. angular accel.:


t 0  t
a  lim
Angular Acceleration
Units of angular acceleration are rad/s²
Every point of the object has the same angular
speed and the same angular acceleration
 The sign of the acceleration does not have to be
the same as the sign of the angular speed
 Angular acceleration is positive if an object
rotating counterclockwise is speeding up or if an
object rotating clockwise is slowing down.
 The instantaneous angular acceleration is
defined as the limit of the average acceleration
as the time interval approaches zero



 Since
rotational motion is analogous to
translational motion we can apply
equations of motion similar to linear motion
with constant acceleration
 Rotational Kinematics
Notes about angular kinematics:
When a rigid object rotates about a fixed axis,
every portion of the object has the same
angular speed and the same angular
acceleration
 i.e. ,, and a are not dependent upon r,
distance form hub or axis of rotation

Example:
A wheel rotates with a constant angular acceleration of
3.50 rad/s2. If the angular velocity of the wheel is 2.00 rad/s at
t=0,
a) Through what angle does the wheel rotate between t=0 and
2.00 s? (in radians and revolutions)
b) What is the angular velocity of the wheel at t = 2.00s?
Want to consider how
angular quantities
relate to linear
quantities
 Consider an arbitrarily
shaped object
 Recall s = rθ
 Determine change in
θ with respect to time


Starting with,
Δθ = Δs / r
Divide by Δt, we arrive at
vt = rω
Where vt is the tangential speed

Similarly, applying the same procedure
as before to the tangential speed, we
get
at = rα
where at is the tangential acceleration
Angular quantities vs. Linear quantities on a
rotating object

Every point on the rotating object has the
same angular motion
 Same for all r

Every point on the rotating object does NOT
have the same linear motion
 Increases with increasing r
Relationship Between Angular
and Linear Quantities

Displacements

Speeds

Accelerations
s  r
ConcepTest
A ladybug sits at the outer edge of a merry-go-round, and
a gentleman bug sits halfway between her and the axis of
rotation. The merry-go-round makes a complete revolution
once each second.The gentleman bug’s angular speed is
1.
2.
3.
4.
half the ladybug’s.
the same as the ladybug’s.
twice the ladybug’s.
impossible to determine
ConcepTest 2
A ladybug sits at the outer edge of a merry-go-round, and
a gentleman bug sits halfway between her and the axis of
rotation. The merry-go-round makes a complete revolution
once each second.The gentleman bug’s angular speed is
1.
2.
3.
4.
half the ladybug’s.
the same as the ladybug’s.
twice the ladybug’s.
impossible to determine
Note: both insects have an angular speed of 1 rev/s

The carnival ride, The
Gravitron, spins at 40
revolutions per minute
and has a radius of 3
meters. If the distance
traveled in one rotation is
the circumference of the
ride, what is your
tangential (linear) speed
if you are sitting on the
outside edge of this ride?
Vector Nature of Angular
Quantities
As in the linear case,
displacement, velocity
and acceleration are
vectors:
 Assign a positive or
negative direction
 A more complete way is
by using the right hand
rule

› Grasp the axis of rotation
with your right hand
› Wrap your fingers in the
direction of rotation
› Your thumb points in the
direction of ω

Angular quantities as
vectors
› Instinctively, we expect
that something be
moving along the
direction of a vector
› This is NOT the case for
angular quantities
› Instead, something is
rotating around the
direction of the vector
› In the world of rotation,
a vector defines an
axis of rotation

Velocity Directions
› In a, the disk rotates
clockwise, the velocity is into
the page
› In b, the disk rotates
counterclockwise, the
velocity is out of the page

Acceleration Directions
› If the angular acceleration and the angular
velocity are in the same direction, the
angular speed will increase with time
› If the angular acceleration and the angular
velocity are in opposite directions, the
angular speed will decrease with time

Dynamics of a Rigid Body
› Previous chapters discussed whether a
force was applied or not, not where it
was applied
› We can generalize concepts of force to
describe rotational dynamics
 Generalize Newton’s Laws
 Equilibrium
› Extend conservation laws
 Energy
 Momentum
Torque

Consider force
required to open door.
Is it easier to open the
door by
pushing/pulling away
from hinge or close to
hinge?
Farther from
from hinge,
larger
rotational
effect!
close to hinge
away from
hinge
 Torque
 Consider
a door
that rotates about
a hinge
› The door is free to
rotate about an
axis through O
› Force is
perpendicular
› τ = rF
› + CCW
- CW
 If
the applied force
is not
perpendicular we
must take the
components of the
Force vector
 However,
only a
net torque
perpendicular will
cause it to rotate
The magnitude of the torque t exerted by
the force F is
τ = rFsinθ
sin 90 = 1, perpendicular max force
 sin 270 = -1, negative dir. max force
 sin 0 = sin 180 = 0, no force

The value of τ depends on the chosen
axis of rotation

Once the axis is chosen, apply right hand rule
to determine direction
1. Point fingers in direction of r
2. Curl fingers in the direction of F
3. Your thumb points in the direction of the
torque
›
Torque is out of the screen

1.
2.

Newton’s Law Analog
The rate of an object does not change, unless
acted on by a net torque
The angular acceleration of an object is
proportional to the net torque
There are three factors that determine the
effectiveness of torque:
›
The magnitude of the force
›
The position of the application of the force
›
The angle at which the force is applied
 If
the net torque is zero, the object’s
rate of rotation doesn’t change
 Equilibrium – arbitrary axis
› May have convenient location
› When solving a problem, you must
specify an axis of rotation and
maintain it
ConcepTest
You are using a wrench and trying to loosen a rusty nut.
Which of the arrangements shown is most effective in
loosening the nut? List in order of descending efficiency the
following arrangements:
ConcepTest
You are using a wrench and trying to loosen a rusty nut.
Which of the arrangements shown is most effective in
loosening the nut? List in order of descending efficiency the
following arrangements:
2, 1, 4, 3
or
2, 4, 1, 3
 Example:
Net Torque

The net torque is the sum of all the
torques produced by all the forces
› Remember to account for the direction of
the tendency for rotation
 Counterclockwise torques are positive
 Clockwise torques are negative
What if two or more different forces
act on lever arm?
Example 1:
N
Determine the net torque:
4m
2m
Given:
weights: w1= 500 N
w2 = 800 N
lever arms: d1=4 m
d2=2 m
500 N
800 N
1. Draw all applicable forces
2. Consider CCW rotation to be positive
Find:
St = ?
t  (500 N )(4 m)  ()(800 N )(2 m)
 2000 N  m  1600 N  m
 400 N  m
Rotation would be CCW

Torque and Equilibrium

First Condition of Equilibrium
 The net external force must be zero
SF  0
S F x  0 an d S F y  0
›
›

This is a necessary, but not sufficient, condition to
ensure that an object is in complete mechanical
equilibrium
This is a statement of translational equilibrium
Second Condition of Equilibrium
 The net external torque must be zero
S t  0
 This is a statement of rotational equilibrium
Axis of Rotation


So far we have chosen obvious axis of rotation
If the object is in equilibrium, it does not
matter where you put the axis of rotation for
calculating the net torque
› The location of the axis of rotation is completely
arbitrary
› Often the nature of the problem will suggest a
convenient location for the axis
› When solving a problem, you must specify an axis
of rotation
 Once you have chosen an axis, you must maintain
that choice consistently throughout the problem
Equilibrium, once again

A zero net torque does not mean the
absence of rotational motion
› An object that rotates at uniform angular
velocity can be under the influence of a
zero net torque
 This is analogous to the translational situation
where a zero net force does not mean the
object is not in motion
Example of a
Free Body Diagram

Isolate the object to
be analyzed

Draw the free body
diagram for that
object
› Include all the external
forces acting on the
object
Where would the 500 N person have to
be relative to fulcrum for zero torque?
Example:
N’
d2 m
y
2m
Given:
weights: w1= 500 N
w2 = 800 N
lever arms: d1=4 m
St = 0
Find:
d2 = ?
500 N
800 N
1. Draw all applicable forces and moment arms
t
t
RHS
  (800 N )(2 m)
LHS
 (500 N )(d 2 m)
800  2 [ N  m]  500  d 2 [ N  m]  0
According to our understanding of torque there
would be no rotation and no motion!
What does it say about acceleration and force?
Thus, according to 2nd Newton’s law SF=0 and a=0!
 d 2  3.2 m

So far: net torque was zero.
What if it is not?
Torque, t, is the tendency of a force to
rotate an object about some axis
 Forces cause acceleration
a = F/m = Δv / Δt
 Torques cause angular accelerations

› Angular acceleration about some fixed point O
at some length r
 Force
and torque must be related in
some way
Torque and Angular
Acceleration


When a rigid object is
subject to a net torque
(≠0), it undergoes an
angular acceleration
The angular
acceleration is directly
proportional to the net
torque
› The relationship is
analogous to ∑F = ma
 Newton’s Second Law
Newton’s Second Law for a
Rotating Object
The angular acceleration is directly proportional to the net
torque
 The angular acceleration is inversely proportional to the
moment of inertia of the object

S t  Ia
There is a major difference between moment of inertia and
mass: the moment of inertia depends on the quantity of matter
and its distribution in the rigid object.
 The moment of inertia also depends upon the location of the
axis of rotation

Moment of Inertia is rotational equivalent of
mass
 Objects with larger mass (more inertia) are
harder to accelerate, objects with larger
moments of inertia are harder to rotate

› Easier to spin a wheel or rod with mass located
at center

Objects moment of inertia depends on not
only on the object’s mass but how the mass
is distributed

Moment of Inertia for particles
m1
r1
r2
r3

m2
m3
I = Σmr2 = m1r12+ m2r22+m3r32
Moment of Inertia of a Uniform Ring
• Calculate the
moment of inertia
• Imagine the hoop is
divided into a
number of small
segments, m1 …
• These segments are
equidistant from the
axis
2
I  S m iri  M R
2
Other
moments
of inertia
The rotational inertia of
an object depends not
only on its mass
distribution but also the
location of the axis of
rotation – compare (f)
and (g), for example.
Two spheres have the same radius and
equal masses. One is made of solid
aluminum, and the other is made from a
hollow shell of gold.
a) solid aluminum
b) hollow gold
c) same
Which one has the bigger moment of
inertia about an axis through its center?
hollow
solid
same mass & radius
Two spheres have the same radius and
equal masses. One is made of solid
aluminum, and the other is made from a
hollow shell of gold.
a) solid aluminum
b) hollow gold
c) same
Which one has the bigger moment of
inertia about an axis through its center?
Moment of inertia depends on
mass and distance from axis
squared. It is bigger for the shell
since its mass is located farther
from the center.
hollow
solid
same mass & radius
Total Energy of Rotating System
An object rotating about some axis with an angular
speed, ω, has rotational kinetic energy ½Iω2
 Energy concepts can be useful for simplifying the
analysis of rotational motion


Conservation of Mechanical Energy
(K E t  K E r  P E g )i  (K E t  K E r  P E g )
› Remember, this is for conservative forces, no
dissipative forces such as friction can be present
› Rolling race!
f
Rotational Kinetic Energy
When using conservation of energy, both
rotational and translational kinetic energy must
be taken into account.
All these objects have the same potential energy at the
top, but the time it takes them to get down the incline
depends on how much rotational inertia they have.
Other
moments
of inertia
The rotational inertia of
an object depends not
only on its mass
distribution but also the
location of the axis of
rotation – compare (f)
and (g), for example.
Work-Energy in a Rotating System
 In the case where there are dissipative
forces such as friction, use the
generalized Work-Energy Theorem
instead of Conservation of Energy

Wnc = KEt + KER + PE
Note on problem solving:


The same basic techniques that were
used in linear motion can be applied to
rotational motion.
› Analogies: F becomes , m becomes I
and a becomes , v becomes ω and
x becomes θ
Techniques for conservation of energy
are the same as for linear systems, as
long as you include the rotational
kinetic energy
 Example:
Problem
Use conservation of
energy to determine the
angular speed of the spool
after the bucket has fallen
4.00m starting from rest.
The light string attached to
the bucket is wrapped
around the spool and
does not slip as it unwinds.
 Similarly
to the relationship between
force and momentum in a linear
system, we can show the relationship
between torque and angular
momentum
 Angular momentum is defined as
L=Iω
› and
L
St 
t
Angular Momentum
Similarly to the relationship between force and momentum in
a linear system, we can show the relationship between
torque and angular momentum
 Angular momentum is defined as L = I ω

L
t
t
(compare to
p
F
t
)
If the net torque is zero, the angular momentum remains
constant
 Conservation of Linear Momentum states: The angular
momentum of a system is conserved when the net external
torque acting on the systems is zero.

› That is, when
St  0, Li  L
f
o r I i i  I f 
f

In an isolated system, the following
quantities are conserved:
› Mechanical energy
› Linear momentum
› Angular momentum
 With
hands and
feet drawn closer
to the body, the
skater’s angular
speed increases
› L is conserved, I
decreases, 
increases
› Ice skater

Example: