Up until now, we have been studying the motion
of point-like objects.
In this module, we are going to consider
the motion of extended objects, specifically
objects that are rigid bodies.
By rigid, we mean that the distance
between any two points on the object always stays the same.
In other words, the shape of a rigid body
does not change or deform.
There is a useful relation known as Chasles' theorem that
states that the general arbitrarily
complicated displacement of a rigid body
can be broken up into two parts-- first,
the translational motion of its center of mass,
and second, the pure rotation of the object
about its center of mass.
We already know how to analyze the center
of mass translational motion through our study
of point-like objects.
This week, we will consider the rotational motion
of rigid bodies, once again distinguishing
between kinematics, a geometric description of the motion,
and dynamics, the underlying cause of changes
in the rotational motion.
On a microscopic level, we can think of an extended rigid body
as made up of a very large number
of small point-like pieces all attached together.
For rotation about a fixed axis through the center of mass,
each little piece will move in a circle
about the rotational axis.
The radius of the circle will vary
depending upon where in the object,
that particular piece sits.
However, because all of these small pieces
are moving collectively as part of a single rigid body,
they will all have the same angular velocity and angular
acceleration at any given instant.
This gives us a way to specify the kinematics or geometrical
description of rotational motion using
the vector quantities of angular velocity and angular
acceleration.
We will also consider the dynamics of rotational motion.
We will see that an applied external force
is able to change the rotational motion of a rigid body,
depending upon exactly where on the body the force is applied.
This leads directly to the concept of torque,
a vector quantity that represents
a sort of rotational force.
Finally, we will see that we can link the kinematics
and dynamics of rotational motion
through a rotational equation of motion, analogous to the role
that Newton's second law plays for translational motion.
This relation states that the angular acceleration
is proportional to the applied torque
and also depends upon how the mass of the object
is distributed.