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Chapter 9
Rotational Dynamics
9.1 The Action of Forces and Torques on Rigid Objects
In pure translational motion, all points on an
object travel on parallel paths.
The most general motion is a combination of
translation and rotation.
9.1 The Action of Forces and Torques on Rigid Objects
According to Newton’s second law, a net force causes an
object to have an acceleration.
What causes an object to have an angular acceleration?
TORQUE
9.1 The Action of Forces and Torques on Rigid Objects
The amount of torque depends on where and in what direction the
force is applied, as well as the location of the axis of rotation.
9.1 The Action of Forces and Torques on Rigid Objects
DEFINITION OF TORQUE
Magnitude of Torque = (Magnitude of the force) x (Lever arm)
  F
Direction: The torque is positive when the force tends to produce a
counterclockwise rotation about the axis.
SI Unit of Torque: newton x meter (N·m)
9.2 Rigid Objects in Equilibrium
If a rigid body is in equilibrium, neither its translational motion nor its
rotational motion changes.
ax  a y  0
F
x
0
 0
F
y
0
  0
9.2 Rigid Objects in Equilibrium
EQUILIBRIUM OF A RIGID BODY
A rigid body is in equilibrium if it has zero translational
acceleration and zero angular acceleration. In equilibrium,
the sum of the externally applied forces is zero, and the
sum of the externally applied torques is zero.
F
x
0
F
y
0
  0
9.2 Rigid Objects in Equilibrium
Example 3 A Diving Board
A woman whose weight is 530 N is
poised at the right end of a diving board
with length 3.90 m. The board has
negligible weight and is supported by
a fulcrum 1.40 m away from the left
end.
Find the forces that the bolt and the
fulcrum exert on the board.
9.2 Rigid Objects in Equilibrium
  F 
2 2
 W W  0
W W
F2 
2
F2

530 N 3.90 m 

 1480 N
1.40 m
9.2 Rigid Objects in Equilibrium
F
y
 F1  F2  W  0
 F1  1480 N  530 N  0
F1  950 N
9.3 Center of Gravity
DEFINITION OF CENTER OF GRAVITY
The center of gravity of a rigid
body is the point at which
its weight can be considered
to act when the torque due
to the weight is being calculated.
9.3 Center of Gravity
W1 x1  W2 x2  
xcg 
W1  W2  
9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis
FT  maT
  FT r
aT  r
  mr 
2
Moment of Inertia, I
9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis
   mr 
2
Net external
torque
Moment of
inertia
 1  m1r12 
 2  m r 
2
2 2

 N  mN rN2 
9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis
ROTATIONAL ANALOG OF NEWTON’S SECOND LAW FOR
A RIGID BODY ROTATING ABOUT A FIXED AXIS
 Moment of   Angular

  

Net external torque  
 inertia
  accelerati on 
  I 
Requirement: Angular acceleration
must be expressed in radians/s2.
 
I   mr 2
9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis
Example 9 The Moment of Inertia Depends on Where
the Axis Is.
Two particles each have mass and are fixed at the
ends of a thin rigid rod. The length of the rod is L.
Find the moment of inertia when this object
rotates relative to an axis that is
perpendicular to the rod at
(a) one end and (b) the center.
9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis
(a)
I   mr 2   m1r12  m2 r22  m0  mL 
2
m1  m2  m
I  mL
2
2
r1  0 r2  L
9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis
 
(b) I   mr 2  m1r12  m2 r22  mL 2 2  mL 2 2
m1  m2  m
I  mL
1
2
2
r1  L 2 r2  L 2
9.4 Newton’s Second Law for Rotational Motion About a Fixed Axis
9.5 Rotational Work and Energy
s  r
W  Fs  Fr
  Fr
W  
9.5 Rotational Work and Energy
DEFINITION OF ROTATIONAL WORK
The rotational work done by a constant torque in
turning an object through an angle is
WR  
Requirement: The angle must
be expressed in radians.
SI Unit of Rotational Work: joule (J)
9.5 Rotational Work and Energy
KE  12 mvT2  12 mr 2 2
vT  r

  mr 
KE   12 mr 2 2  12
2
2
 12 I 2
9.5 Rotational Work and Energy
DEFINITION OF ROTATIONAL KINETIC ENERGY
The rotational kinetic energy of a rigid rotating object is
KER  12 I 2
Requirement: The angular speed must
be expressed in rad/s.
SI Unit of Rotational Kinetic Energy: joule (J)
9.6 Angular Momentum
DEFINITION OF ANGULAR MOMENTUM
The angular momentum L of a body rotating about a
fixed axis is the product of the body’s moment of
inertia and its angular velocity with respect to that
axis:
L  I
Requirement: The angular speed must
be expressed in rad/s.
SI Unit of Angular Momentum: kg·m2/s
9.6 Angular Momentum
PRINCIPLE OF CONSERVATION OF ANGULAR MOMENTUM
The angular momentum of a system remains constant (is
conserved) if the net external torque acting on the system
is zero.
9.6 Angular Momentum
Conceptual Example 14 A Spinning Skater
An ice skater is spinning with both
arms and a leg outstretched. She
pulls her arms and leg inward and
her spinning motion changes
dramatically.
Use the principle of conservation
of angular momentum to explain
how and why her spinning motion
changes.
Problems to be solved
• 9.6, 9.12, 9.14, 9.21, 9.25, 9.40, 9.51,
9.61, 9.69, 9.74
• B9.1 A bicycle wheel has a mass of 2kg
and a radius of 0.35m. What is its
moment of inertia? Ans: 0.245kgm2
• B9.2 A grinding wheel, a disk of uniform
thickness, has a radius of 0.08m and a
mass of 2kg. (a) What is its moment of
inertia? (b) How large a torque is needed
to accelerate it from rest to 120rad/s in 8s?
Ans: (a) 0.0064kgm2 (b) 0.096Nm
• B9.3 A student holding a rod by the centre
subjects it to a torque of 1.4Nm about an
axis perpendicular to the rod, turning it
through 1.3rad in 0.75s. When the student
holds the rod by the end applies the same
torque to the rod, through how many
radians will the rod turn in 1.0s?
Ans: 0.58rad
• B9.4 An ice skater starts spinning at a rate
of 1.5rev/s with arms extended. She then
pulls her arms in close to her body,
resulting in a decrease of her moment of
inertia to three quarters of the initial value.
What is the skater’s final angular velocity?
Ans: 2rev/s