Download 08_Rotational Motion and Equilibrium

Lecture Outline
Chapter 8
College Physics, 7th Edition
Wilson / Buffa / Lou
© 2010 Pearson Education, Inc.
Chapter 8
Rotational Motion and
Equilibrium
© 2010 Pearson Education, Inc.
Units of Chapter 8
Rigid Bodies, Translations, and Rotations
Torque, Equilibrium, and Stability
Rotational Dynamics
Rotational Work and Kinetic Energy
Angular Momentum
© 2010 Pearson Education, Inc.
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 an exception!
© 2010 Pearson Education, Inc.
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.
© 2010 Pearson Education, Inc.
8.1 Rigid Bodies, Translations,
and Rotations
For an object that is rolling without slipping,
© 2010 Pearson Education, Inc.
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.
© 2010 Pearson Education, Inc.
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.
© 2010 Pearson Education, Inc.
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.
© 2010 Pearson Education, Inc.
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.
© 2010 Pearson Education, Inc.
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.
© 2010 Pearson Education, Inc.
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.
© 2010 Pearson Education, Inc.
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.
© 2010 Pearson Education, Inc.
8.2 Torque, Equilibrium, and Stability
Whether equilibrium is stable or unstable
depends on the width of the base of support.
© 2010 Pearson Education, Inc.
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.
© 2010 Pearson Education, Inc.
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.
© 2010 Pearson Education, Inc.
8.3 Rotational Dynamics
The moments of inertia of certain symmetrical
shapes can be calculated. Here is a sample:
© 2010 Pearson Education, Inc.
8.3 Rotational Dynamics
The parallel-axis theorem relates the moment
of inertia about any axis to the moment of
inertia about a parallel axis that goes through
the center of mass.
© 2010 Pearson Education, Inc.
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:
© 2010 Pearson Education, Inc.
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.
© 2010 Pearson Education, Inc.
8.4 Rotational Work and Kinetic Energy
There is a strict analogy between linear and
rotational dynamic quantities.
© 2010 Pearson Education, Inc.
8.5 Angular Momentum
Definition of angular momentum:
In vector form (the direction is again
given by the right-hand rule):
© 2010 Pearson Education, Inc.
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).
© 2010 Pearson Education, Inc.
8.5 Angular Momentum
Internal forces can change a system’s
moment of inertia; its angular speed will
change as well.
© 2010 Pearson Education, Inc.
8.5 Angular Momentum
This can be demonstrated in the classroom.
(What purpose do the hand weights serve?)
© 2010 Pearson Education, Inc.
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.
© 2010 Pearson Education, Inc.
8.5 Angular Momentum
An external torque on a rotating object causes
it to precess.
© 2010 Pearson Education, Inc.
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.
© 2010 Pearson Education, Inc.
Summary of Chapter 8
Moment of inertia:
Newton’s second law:
Parallel-axis theorem:
Rotational work, power, and kinetic energy:
© 2010 Pearson Education, Inc.
Summary of Chapter 8
Angular momentum:
Newton’s second law, again:
In the absence of an external net torque,
angular momentum is conserved.
© 2010 Pearson Education, Inc.