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With the subject of motion in more than one dimension broached, we can
consider things other than simple linear motion. For example, many
types of motion involve curved paths rather than linear ones. We have
already seen one example in the parabolic shape of the trajectory of an
object in free-fall.
Another example is simply the motion of an object in a circular path.
This includes motions as diverse as cars on a Ferris wheel to satellites
in orbit around the earth. This kind of motion is somewhat special in that
the object of interest maintains a constant speed as it goes around the
circle, i.e. the magnitude of its velocity doesn't change. However, the
direction of the velocity does have to change continuously to maintain
the circular path. Whenever the velocity magnitude, direction, or both
is changed, there must be a non-zero acceleration acting to change the motion.
Look at the figure below.
We've already discussed radians and their
connection to radii and arc-lengths. Let's review how they would apply
to finding the acceleration of an object in a circular path by
looking at the figure below.
The object moves in a circular path of radius R. At some time,
t_0,
the velocity has direction shown in the figure and magnitude v.
Some short time later, at t_1,
the object has a velocity with a new direction but the same old magnitude,
v. The angular position of the object has changed by an amount
.
The velocity change is, as shown,
.
The distance traveled by the
object in time 
is the arc-length 
.
We can now start to relate the items we have defined. First, notice that,
by the definition of radians (see your
lecture notes from the first week), we have
where we have only talked about magnitudes. The direction for
we get by noting that it is approximately perpendicular to
by our picture.
Since both
and
are
equal to ,
then they are equal to each other, so
The definition of average acceleration during the time
is
, so
where we remember that the magnitude of the objects velocity, v,
and the radius R are constant in time. The time taken for the
object to travel arc-length
is set by the time difference and the velocity, so we know that
.
Therefore, our average acceleration is just
a ~ v/R * v = v² /R
As usual, we move from the average acceleration (which is just the average
rate of change of the velocity) to the exact instantaneous acceleration
by taking the limit as
.
This is the same as letting
.
In that case, the change in velocity is
exactly perpendicular to
. But since
is tangent to
the circular path and therefore perpendicular to R, then the change in
velocity is parallel or anti-parallel to R.
From our picture, we see
that 
is anti-parallel (it points in toward the center while
the radius points outward). Therefore, the acceleration needed to maintain
a perfectly circular path of radius R
with constant speed is
a = v² /R
with direction inward toward the center of the circular path. When such
a condition prevails, we say that the object is subject to a
centripetal acceleration and that the object is undergoing
uniform circular motion.
Of course, for an acceleration to be acting, there must be a force. Any
force that provides an acceleration satisfying the centripetal condition
(i.e. center-pointing, magnitude constant at
) is a centripetal
force. A string tied to a rock whirled at constant velocity in a horizontal
circle provides a centripetal force through the tension supplied by your
hand. For the Ferris wheel, the tension is provided by
struts which connect the seats to the wheel and the wheel to its center.
Satellites are maintained in orbit by earth's gravity.
For another interesting example where knowledge of circular motion is
important, click the image below.
Practice using the concepts of centripetal acceleration on
a few, short exercises,
4-5 and
4-6.
Next: Connection between circular
Up: Completing the Circle
Previous: Vectors