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Chapter 8
Rotational Motion and
Equilibrium
Units of Chapter 8
Rigid Bodies, Translations, and Rotations
Torque, Equilibrium, and Stability
Rotational Dynamics
Rotational Work and Kinetic Energy
Angular Momentum
8.1 Rigid Bodies, Translations,
and Rotations
A rigid body is an object or a system of particles in
which the distances between particles are fixed (remain
constant).
In other words, a rigid body must be solid (but not
all solid bodies are rigid).
Raise your hand if you can think of solid body that
is not rigid.
8.1 Rigid Bodies, Translations,
and Rotations
A rigid body may have translational motion,
rotational motion, or a combination. It may roll with
or without slipping.
8.1 Rigid Bodies, Translations,
and Rotations
For an object that is rolling without slipping,
8.2 Torque, Equilibrium, and Stability
It takes a force to start an object rotating; that force
is more effective the farther it is from the axis
of rotation, and the closer it is to being
perpendicular to the line to that axis.
8.2 Torque, Equilibrium, and Stability
The perpendicular distance from the line of force
to the axis of rotation is called the lever arm.
The product of the force and the lever arm is
called the torque.
8.2 Torque, Equilibrium, and Stability
Torque is a vector (it produces an angular
acceleration), and its direction is along the axis of
rotation, with the sign given by the right-hand rule.
8.2 Using Torque
From Walmart’s website… no one has
ever been this happy to change a tire,
don’t be fooled…
8.2 Torque, Equilibrium, and Stability
In order for an object to be in equilibrium, the net
force on it must be zero, and the net torque on it
must be zero as well.
8.2 Torque, Equilibrium, and Stability
The left stick and the triangle are in equilibrium;
they will neither translate nor rotate. The stick on
the right has no net force on it, so its center of
mass will not move; the torque on it is not zero, so
it will rotate.
8.2 Torque, Equilibrium, and Stability
For an object to be
stable, there must be no
net torque on it around
any axis. The axis used
in calculation may be
chosen for convenience
when there is no motion.
Stable: τnet = 0 m·N
8.2 Torque, Equilibrium, and Stability
What is the unknown
mass 3 to the right?
τ = F1r1 + F2r2 – F3r3 = 0 mN
τ = m1gr1 + m2gr2 – m3gr3 = 0 mN
m3 = ?
8.2 Torque, Equilibrium, and Stability
If an object is in stable
equilibrium, any
displacement from the
equilibrium position
will create a torque
that tends to restore
the object to
equilibrium. Otherwise
the equilibrium is
unstable.
8.2 Torque, Equilibrium, and Stability
Whether equilibrium is stable or unstable
depends on the width of the base of support.
Torque Questions
• Book Pages 289-290
– #’s 26-28, 37-38, 40, 43
8.3 Rotational Dynamics
The net torque on an object causes its angular
acceleration. For a point particle, the relationship
between the torque, the force, and the angular
acceleration is relatively simple.
8.3 Rotational Dynamics
We can consider an extended object to be a lot of
near-point objects stuck together. Then the net
torque is:
The quantity inside the parentheses is called the
moment of inertia, I.
8.3 Rotational Dynamics
The moments of inertia of certain symmetrical
shapes can be calculated. Here is a sample:
8.4 Rotational Work and Kinetic Energy
The work done by a torque:
As usual, the power is the rate
at which work is done:
8.4 Rotational Work and Kinetic Energy
The work–energy theorem still holds—the net
work done is equal to the change in the kinetic
energy. This gives us the form of the rotational
kinetic energy.
8.4 Rotational Work and Kinetic Energy
There is a strict analogy between linear and
rotational dynamic quantities.
8.5 Angular Momentum
Definition of angular momentum:
In vector form (the direction is again given
by the right-hand rule):
8.5 Angular Momentum
The rate of change of the angular momentum is
the net torque:
Angular momentum is conserved:
In the absence of an external, unbalanced torque, the
total (vector) angular momentum of a system is
conserved (remains constant).
8.5 Angular Momentum
Internal forces can change a system’s moment of
inertia; its angular speed will change as well.
8.5 Angular Momentum
The conservation of
angular momentum
means its direction
cannot change in the
absence of an
external torque. This
gives spinning objects
remarkable stability.
8.5 Angular Momentum
This can be demonstrated in the classroom.
(What purpose do the hand weights serve?)
8.5 Angular Momentum
An external torque on a rotating object causes it to
precess.
Summary of Chapter 8
Pure translational motion: all parts of object have
same velocity
Pure rotational motion: center of mass does not
move; all parts of object have same rotational
velocity
Rolling without slipping:
Torque:
An object in mechanical equilibrium has no net
force and no net torque acting on it.
Summary of Chapter 8
Moment of inertia:
Newton’s second law:
Parallel-axis theorem:
Rotational work, power, and kinetic energy:
Summary of Chapter 8
Angular momentum:
Newton’s second law, again:
In the absence of an external net torque,
angular momentum is conserved.
Chapter 8 Review
Pages 288 – 295:
Questions:
• 8.3: Rotational Dynamics
• 8.4: Rotational Work / KE
• 8.5: Angular Momentum
59-60
82-83
101,105,109,110
8.1 Rigid Bodies, Translations,
and Rotations
A rigid body may have translational motion,
rotational motion, or a combination. It may roll
with or without slipping.
8.2 Torque, Equilibrium, and Stability
Torque is a vector (it produces an angular
acceleration), and its direction is along the axis of
rotation, with the sign given by the right-hand rule.
8.2 Torque, Equilibrium, and Stability
If an object is in stable
equilibrium, any
displacement from the
equilibrium position
will create a torque
that tends to restore
the object to
equilibrium. Otherwise
the equilibrium is
unstable.
8.2 Torque, Equilibrium, and Stability
What is the unknown
mass 3 to the right?
τ = F1r1 + F2r2 – F3r3 = 0 mN
τ = m1gr1 + m2gr2 – m3gr3 = 0 mN
m3 = ?
8.5 Angular Momentum