There are a few basic physical concepts that are fundamental
to a proper understanding of rotational motion. With a steady grasp
of these concepts, you should encounter no major difficulties in
making the transition between the mechanics of translational motion
and of rotational motion.
The questions on rotational motion on SAT II Physics deal
only with rigid bodies. A rigid body is an object that
retains its overall shape, meaning that the particles that make
up the rigid body stay in the same position relative to one another.
A pool ball is one example of a rigid body since the shape of the
ball is constant as it rolls and spins. A wheel, a record, and a
top are other examples of rigid bodies that commonly appear in questions
involving rotational motion. By contrast, a slinky is not a rigid
body, because its coils expand, contract, and bend, so that its
motion would be considerably more difficult to predict if you were
to spin it about.
Center of Mass
The center of mass of an object, in case
you have forgotten, is the point about which all the matter in the
object is evenly distributed. A net force acting on the object will
accelerate it in just the same way as if all the mass of the object
were concentrated in its center of mass. We looked at the concept
of center of mass in the previous chapter’s discussion of linear momentum.
The concept of center of mass will play an even more central role
in this chapter, as rotational motion is essentially defined as
the rotation of a body about its center of mass.
Axis of Rotation
The rotational motion of a rigid body occurs when every
point in the body moves in a circular path around a line called
the axis of rotation, which cuts through the center
of mass. One familiar example of rotational motion is that of a
spinning wheel. In the figure at right, we see a wheel rotating
counterclockwise around an axis labeled O that
is perpendicular to the page.
As the wheel rotates, every point in the rigid body makes
a circle around the axis of rotation, O.
We’re all very used to measuring angles in degrees, and
know perfectly well that there are 360º in a circle, 90º
in a right angle, and so on. You’ve probably noticed that 360 is
also a convenient number because so many other numbers divide into
it. However, this is a totally arbitrary system that has its origins
in the Ancient Egyptian calendar which was based on a 360-day
It makes far more mathematical sense to measure angles
in radians (rad). If we were to measure the arc of
a circle that has the same length as the radius of that circle,
then one radian would be the angle made by two radii drawn to either
end of the arc.
Converting between Degrees and Radians
It is unlikely that SAT II Physics will specifically ask
you to convert between degrees and radians, but it will save you
time and headaches if you can make this conversion quickly and easily. Just
remember this formula:
You’ll quickly get used to working in radians, but below
is a conversion table for the more commonly occurring angles.
|Value in degrees
||Value in radians
Calculating the Length of an Arc
The advantage of using radians instead of degrees, as
will quickly become apparent, is that the radian is based on the
nature of angles and circles themselves, rather than on the arbitrary
fact of how long it takes our Earth to circle the sun.
For example, calculating the length of any arc in a circle
is much easier with radians than with degrees. We know that the
circumference of a circle is given by P = 2πr
and we know that there are 2π
radians in a circle.
If we wanted to know the length, l
of the arc described by any angle
, we would know that this arc is a fraction
of the perimeter, (/2π)P
. Because P
, the length of the arc would be: