


Rotational Kinematics
You are now going to fall in love with the word angular.
You’ll find that for every term in kinematics that you’re familiar
with, there’s an “angular” counterpart: angular displacement, angular
velocity, angular acceleration, etc. And you’ll
find that, “angular” aside, very little changes when dealing with
rotational kinematics.
Angular Position, Displacement, Velocity, and Acceleration
SAT II Physics is unlikely to have any questions that
simply ask you to calculate the angular position, displacement,
velocity, or acceleration of a rotating body. However, these concepts
form the basis of rotational mechanics, and the questions you will encounter
on SAT II Physics will certainly be easier if you’re familiar with
these fundamentals.
Angular Position
By convention, we measure angles in a circle in a counterclockwise
direction from the positive xaxis. The angular
position of a particle is the angle, , made between the line connecting that
particle to the origin, O, and the
positive xaxis, measured counterclockwise. Let’s
take the example of a point P on a
rotating wheel:
In this figure, point P has
an angular position of . Note that every point
on the line has the same angular position: the angular
position of a point does not depend on how far that point is from
the origin, O.
We can relate the angular position of P to
the length of the arc of the circle between P and
the xaxis by means of an easy equation:
In this equation, l is the
length of the arc, and r is the radius
of the circle.
Angular Displacement
Now imagine that the wheel is rotated so that every point
on line moves from an initial angular position
of to a final angular position of . The angular displacement, , of line is:
For example, if you rotate a wheel counterclockwise such
that the angular position of line changes from = 45º = π/4 to = 135º = 3π/4,
as illustrated below, then the angular displacement of line is 90º or π/2 radians.
For line to move in the way described
above, every point along the line must rotate 90º counterclockwise.
By definition, the particles that make up a rigid body must stay
in the same relative position to one another. As a result, the angular
displacement is the same for every point in a rotating rigid body.
Also note that the angular distance a point has rotated
may or may not equal that point’s angular displacement. For example,
if you rotate a record 45º clockwise and then 20º counterclockwise,
the angular displacement of the record is 25º, although
the particles have traveled a total angular distance of 65º.
Hopefully, you’ve already had it hammered into your head that distance
and displacement are not the same thing: well, the same distinction applies
with angular distance and angular displacement.
Angular Velocity
Angular velocity, , is defined as the change in the angular
displacement over time. Average angular velocity, , is defined by:
Angular velocity is typically given in units of rad/s.
As with angular displacement, the angular velocity of every point
on a rotating object is identical.
Angular Acceleration
Angular acceleration, , is defined as the rate of change of angular
velocity over time. Average angular acceleration, , is defined by:
Angular acceleration is typically given in units of rad/s^{2}.
