Phy 201: General Physics I
Chapter 9: Rotational Dynamics
Lecture Notes
Rigid Objects & Torque
Torque is a vector quantity that represents the application of
force to a body resulting in rotation (or change in rotational state)
F

F
r
r




Definition:
= Fr = F  sin r
• The SI units for torque are N.m (not to be confused with joules)
• Torque depends on:
– The Lever arm (leverage), r
– The component of force perpendicular to lever arm, F= F  sin  ˆi
Newton’s 2nd Law
(for Rotational Motion)
When there is a net torque exerted on a rigid body its state
of rotation ( ω ) will change depending on:

1. Amount of net torque, 
2. Rotational inertia of object, I
a =

I
=
1 +  2 + ...
I
1 n
=  i
I i=1
Note: No net torque means a = 0 and w = constant}
Alternatively, the net torque exerted on a body is equal to
the product of the rotational inertia and the angular
acceleration:
n

=  1 +  2 + ... =

i=1
i
= Ia
This is Newton’s 2nd Law (for rotation)!!
Rigid Objects in Equilibrium
A rigid body is in equilibrium when:
– No net force  no change in state of motion
(v = constant)
F = F
x
+ Fy + Fz = 0
– No net torque  no change in state of rotation
(w = constant)

=0
Both of the above conditions must be met for an object to be
in mechanical equilibrium!
Note: an object can be both moving and rotating and still be
in mechanical equilibrium
Center of Gravity (COG)
The COG is the location where all of an
object’s weight can be considered to act
when calculating torque
• The COG may or may not actually be on
the object (i.e. consider a hollow sphere
or a ring)
• When an object’s weight produces a
torque on itself, it acts at its center of
mass
To calculate center of gravity: n
W1x1 + W2x2 + ...
xcg=
=
W1+ W2+ ...
 Wx
i=1
i
i
Wtot
n
W y + W2y2 + ...
ycg= 1 1
=
W1+ W2+ ...
 Wy
i=1
i
Wtot
i
Moment of Inertia
The resistance of a rigid body to changes in its state of
rotational motion ( ω ) is called the moment of inertia (or
rotational inertia)
The Moment of Inertia (I) depends on:
1. Mass of the object, m
2. The axis of rotation
3. Distribution (position) of mass about the axis of rotation
Definition (For a discrete distribution of mass):
n
I = m1r12 + m2r22 + ... mnrn2 =
Where:
2
mr
 ii
i=1
mi is the mass of a small segment of the object
ri is the distance of the mass mi from the axis of rotation
The SI units for I are kg.m2
Note: For a continuous distribution of mass there are more
sophisticated techniques for calculating the moment of inertia
Moment of Inertia (common examples)
1. A simple pendulum:
R
Ipendulum = mR2
m
2. A thin ring:
Iring= mR2
R
dm
3. A solid cylinder:
dm
R
H
mR 2
Icylinder =
2
Moment of Inertia (common examples, cont.)
4. A hollow sphere:
R
m
2mR2
Ihollow sphere=
3
5. A solid sphere:
2mR 2
Isolid sphere=
5
R
m
6. A thin rod:
L
.
m
mL2
Irod=
12
The Parallel Axis Theorem
• The Parallel Axis Theorem is used to determine the
moment of inertia for a body rotated about an axis a
distance, l, from the center of mass:
I = Icm + mL2
Example: A 0.2 kg cylinder (r=0.1 m) rotated about an
axis located 0.5 m from its center:
I = Icm + mL2
L
mr 2
I=
+ mL2
2
I =  0.002 + 0.050  kg  m2
I = 0.052 kg  m2
Work & Rotational Kinetic Energy
When torque is applied to an object and a rotation is
produced, the torque does work:
W = θ
When there is net torque:
WNet =
θ = Iaθ
Since w2 - wo2 = 2aq
 ω2 -ω2o 
2
2
1
1
WNet = Iaθ = I 
=
Iω
Iω

o
2
2
2


The rotational kinetic energy (Krot) for an object is defined as:
Krot =
1
2
Iω2  WNet =
1
2
Iω2 -
1
2
Iω2o = Krot -Krot o = Krot
This is the Work-Energy Theorem (for rotation)!!
Rolling objects & Inclined Planes
Consider a solid disc & a hoop rolling down a hill (inclined plane):
Solid
disc
Hoop
q
q
1. Apply Newton’s 2nd Law (force) to each object
2. Apply Newton’s 2nd Law (torque) to each object
3. What is the acceleration of each object as they roll down the
hill?
4. Which one reaches the bottom first?
Angular Momentum
• Angular momentum is the rotational analog of linear
momentum
• It represents the “quantity of rotational motion” for an object
(or its inertia in rotation)
• Angular Momentum (a vector we will treat as a scalar) is
defined as:
L = I.w
p
• Note:
Angular Momentum is related to Linear Momentum:
L=
r.p
sin q
r is distance from the axis of rotation
p is the linear momentum
q is the angle between r and p
• SI units of angular momentum are kg.m2/s
q
r
Conservation of Angular Momentum
• When there is no net torque acting on an object (or no net
torque acts on a system) its angular momentum (L) will remain
constant or
L1 = L2 = … = a constant value
Or
I1w1 = I2w2 = etc…
• This principle explains why planets move faster the closer
the get to the sun and slower the they move when they are
farther away
slowest
fastest
Note: this is a re-statement of Newton’s 1st Law
Conservation of Angular Momentum (Examples)
• A figure skater
• A high diver
• Water flowing down a drain
Archimedes (287-212 BC)
• Possibly the greatest mathematician in
history
• Invented an early form of calculus
• Discovered the Principle of Buoyancy (now
called Archimedes’ Principle)
• Introduced the Principle of Leverage
(Torque) and built several machines based
on it.
“Give me a point of
support and I will
move the Earth”