Chapter 9: Rotation of rigid bodies
•  Rotational kinematics
•  Linear VS Angular kinematics
•  It’s really similar!
•  Definition of moments of inertia and determination of
rotational kinetic energy
•  You’ve already done it in HMWK CH 6 (The kinetic
energy of a rotating bar, remember?)
•  Compare mass and moment of inertia
•  Calculation of the moment of inertia
•  Time to practice integral
Introduction
•  “Real-world” rotations can be very
complicated because of stretching
and twisting of the rotating object.
•  We start with the simple one: Rigid
body
•  Rigid body: An object with perfectly
definite and unchanging shape/size
•  Is spinning tennis ball a rigid
body?
A speedometer as our starting model
•  Examples of rotations
in our daily life are
abundant and
numerous:
•  DVD in use
•  Clock, etc
•  A car’s analog
speedometer gives us a
very good example to
begin defining
rotational motion.
Where is the axis of rotation?
College students use radians, not degrees (for kids)
•  One complete cycle of
360° is one revolution.
•  One complete
revolution is 2π
radians.
•  Relating the two, 360°
= 2 π radians or 1
radian = 57.3°.
•  In degrees, what are
1 1 1 1
π? π? π? π?
4
3 2 6
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Angular displacement is the angle being swept out
•  Like a second hand sweeping around a clock, a radius
vector will travel through a displacement of degrees,
radians, or revolutions.
•  We denote angular displacement as Θ (theta). It is the
angular equivalent of x or y in earlier chapters.
Angular velocity
•  The angular velocity is the angle swept out divided by the time it
took to sweep out the angular displacement.
•  Angular velocity is denoted by the symbol ω (omega).
•  Angular velocity is measured in radians per second (SI standard)
as well as other measures such as r.p.m. (revolutions per second).
dθ
ω=
dt
€
€
Example 9.1: flywheel test
θ = (2.0rad /s3 )t 3
r = 0.35m
From t=0 to 5s, what is average
Angular velocity? Instant ω at 5s?
How far did a point on the rim travel?
Angular velocity is a vector: defined by the rotation axis
•  You can visualize the position of the vector by sweeping out the
angle with the fingers of your right hand. The position of your
thumb will be the position of the angular velocity vector. This is
called the “right-hand rule.”
•  Why not “left-hand-rule”
Rotation around multiple axes: Earth rotation
Do you know the origin of seasons on earth?
Is it due to the distance from sun to earth?
Angular acceleration
•  The angular acceleration is the change of angular
velocity divided by the time interval during which the
change occurred.
•  Use the symbol α (alpha) to denote radians2 per second.
dω d θ
α=
= 2
dt
dt
•  Back to example 9.1:
θ = (2.0rad /s3 )t 3
r = 0.35m
€
what is the angular acceleration
€
as a function of time?
Angular acceleration is a vector
•  The angular acceleration vector will be parallel or
antiparallel to the angular velocity vector (as
determined by the RHR).
linear motion VS angular motion with constant acceleration
If you mastered linear motion with constant acceleration,
you are home-free!
If not, now it is your chance to master both at the same time!
Example 9.3: Angular motion of a DVD slowing down
t =0:
ω = 27.5rad /s
α = −10.0rad /s2
What is at the angular velocity ω t=0.3s?
What angle does the line PQ make with the +x axis?
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Linear velocity vs angular velocity and circular motion
•  On a rotating rigid body,
all particles have the same
angular velocity.
What about the linear
velocity?
What is the acceleration of P
if there is no angular
acceleration?
v = ωr
What if α is not 0? a = αr
tan
arad = ω 2 r
Bicycle pedals and gears
•  How are the angular speeds of the two sprockets related to the
number of teeth (and their sizes of) on each sprocket?
Kinetic energy 2.0: Rotational energy
•  A rotating body has
kinetic energy. But
the speeds depends
on the positions
•  Just like linear
kinetic energy is
½ mv2, the angular
energy will be
determined by
½ Iω2.
Moments of inertia
K=
1
1
1
miv i2 = ∑ miω i2 ri2 = (∑ mi ri2 )ω i2
∑
2 i
2 i
2 i
I = ∑ mi ri2
i
1
K = Iω 2
2
Just like mass describes how resistant an object is to acceleration
Moment€of inertia describes how resistant an object is to angular
acceleration
The distribution of mass matters!
The rotation axis matters too!
 2
For a continuous body:
I=
€
∫ ρ(r )r dV
Rotational energy changes if parts shift and I changes
•  Even if the masses are
equal, rearranging the
components of a rotating
system can change the
moment of inertia and the
rotational energy.
•  Example 9.7:
•  A) What is I when axis goes through A
What is K if ω=4.0 rad/s
• 
B) What is I when axis goes through B
and C
Finding the moment of inertia for common shapes
Finding the moment of inertia for common shapes
How to you obtain b) from a)? Hint: use parallel-axis theorem
If a) and b) are both rotating at the same angular velocity, which one has
the larger kinetic energy?
Unwinding cable: A classic example
•  Block (m) is released from rest at hight h. As it comes down, the
cylinder (M, R) will start to rotate. What is the angular speed of
the cylinder as the block hits the floor?
Center of mass and the Parallel Axis Theorem
•  CH8: center of mass
∑m x
=
∑m
i
x cm
P: (a, b)
Center of mass: (xcm, ycm)
i
i
i
i

rcm
∑ m r
=
∑m
i i
i
i
i
•  It’s always easier to €rotate around the
axis trough the center of mass
•  Parallel Axis Theorem:
How?
€
I p = Icm + Md 2
I p = ∑ mi [(x i − a) 2 + (y i − b) 2 ]
•  Example: Knowing
, how to get I=? For
I=? For a rod rotating around any position
•  Instead of using integrals, could you use the
parallel axis theorem?
Or any other methods?
Rotation of a uniform disk about a center axis
How to get (f) from (g)?
Solid cylinder is the sum of a series of thin-walled hollow cylinder
One such segment (from r to r+dr, with mass dm) has dI as
dI = r 2 dm
M(2πr)dr
dm =
πR 2
∴I =
∫ dI =
∫ r2
M(2πr)dr 2M
= 2
2
πR
R
∫ r dr
3
Summary
• Rotational kinematics: Angular velocity, angular acceleration (vector!)
Linear VS angular
kinematics
v = ωr
a = αr
• Moment of inertia: I
I = ∑ mi ri2 ⇒ I =
∫ ρ(r)r dV
2
i
1 2
1
K = mv
⇒ K = Iω 2
€
2
2
• Rotational kinetic energy:
€
When potential energy can be converted
to kinetic energy, which also
Includes Rotational kinetic energy!
€
• Calculating the moment of inertia:
• Application of Parallel Axis Theorem!
• Icm is also the smallest
€
I p = Icm + Md 2