There is a tendency to believe that if an object is moving at constant speed then it has no
acceleration. This is indeed true in the case of an object moving along a straight line
path. On the other hand, a particle moving on a curved path is accelerating whether the
speed is changing or not. Velocity has both magnitude and direction. In the case of a
particle moving on a curved path, the direction of the velocity is continually changing,
and thus the particle has acceleration.
We now turn our attention to the case of an object moving in a circle. We’ll start with the
simplest case of circular motion, the case in which the speed of the object is a constant, a case
referred to as uniform circular motion. For the moment, let’s have you be the object. Imagine
that you are in a car that is traveling counterclockwise, at say 40 mph, as viewed from above,
around a fairly small circular track. You are traveling in a circle. Your velocity is not constant.
The magnitude of your velocity is not changing (constant speed), but the direction of your
velocity is continually changing, you keep turning left! Now if you are continually turning left
then you must be continually acquiring some leftward velocity. In fact, your acceleration has to
be exactly leftward, at right angles to your velocity because, if your speed is not changing, but
dv
your velocity is continually changing, meaning you have some acceleration a =
, then for
dt
every infinitesimal change in clock reading dt, the change in velocity v
d that occurs during that
infinitesimal time interval must be perpendicular to the velocity itself. (If it wasn’t
perpendicular, then the speed would be increasing or decreasing.) So no matter where you are in
the circle (around which you are traveling counterclockwise as viewed from above) you have an
acceleration directed exactly leftward, perpendicular to the direction of your velocity. Now what
is always directly leftward of you if you are traveling counterclockwise around a circle?
Precisely! The center of the circle is always directly leftward of you. Your acceleration is thus,
always, center directed. We call the center-directed acceleration associated with circular motion
centripetal acceleration because the word “centripetal” means “center-directed.” Note that if you
are traveling around the circle clockwise as viewed from above, you are continually turning right
and your acceleration is directed rightward, straight toward the center of the circle. These
considerations apply to any object—an object moving in a circle has centripetal (center-directed)
acceleration.
We have a couple of ways of characterizing the motion of a particle that is moving in a circle.
First, we characterize it in terms of how far the particle has traveled along the circle. If we need
a position variable, we establish a start point on the circle and a positive direction. For instance,
for a circle centered on the origin of an x-y plane we can define the point where the circle
intersects the positive x axis as the start point, and define the direction in which the particle must
move to go counterclockwise around the circle as the positive direction. The name given to this
position variable is s. The position s is the total distance, measured along the circle, that the
ds
particle has traveled. The speed of the particle is then the rate of change of s,
and the
dt
direction of the velocity is tangent to the circle. The circle itself is defined by its radius. The
second method of characterizing the motion of a particle is to describe it in terms of an
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