A mobile moving in a circle with the
magnitude of the velocity constant is say to move in uniform circular motion.
Even when, apparently, such an object has a constant velocity because the
changes in the direction of motion implies the present of an acceleration. This
acceleration is called centripetal acceleration. The next section shows the
meaning and origin of such an acceleration.

Let us consider a mobile moving in a circle at
known constant speed (constant magnitude of the vector velocity), such an
object still undergo acceleration because of the constant change in the
direction of the vector velocity.

The drawing on the left illustrates
such a situation, at the initial time
the point P is at an angle
(initial angle)
with respect to the horizontal axis moving with the velocity
(initial velocity)
in a circle of radius
.
A few instant later, point P has turn to the new angle
(final
angle) and it is moving with the velocity
(final
velocity) .
Uniform circular motion means that the magnitude of the velocity does
not change during the motion,
,
even when the direction of motion is constantly changing
,
.
By definition, a change in velocity implies the presence of an
acceleration,
(definition of
average acceleration),
where the change in time,
,
can be expressed in terms of the distance traveled and the speed of the
point P (or magnitude of the velocity of the point)
.

At this point, the change in time is not in the
limiting case .
Therefore, the acceleration is not instantaneous but rather the average
acceleration. Since the speed of the point is constant, the
instantaneous speed is the same as the average speed (this is not true
for the velocity in this case). Thus, the equation relating the distance
travel, the speed, and the time is accurate independent of the
time
being a
finite time or an infinitely small time, case
. On the other side, the distance traveled by point P corresponds
to the arc length subtended by the angle obtained from the difference
between the two angles
and
;
that is,
(these two angles can not be measured in degrees but rather in radians,
see below). Moreover, for a given angle,
the arc length corresponds to the product between the angle and the
radius of the circle,
.
Therefore, the change in time is
and the corresponding acceleration is
. In order to obtain the acceleration, still the magnitude and direction
of the ratio
needs to be calculated. Algebraically, the vector subtraction of vectors
and
is obtain solving the addition of vectors
and
as represented by
.
In order to complete the operation, let us place the final vector
velocity at the point of interception between the sustain lines for the
initial and final vector velocities
.
Next, let us place the tail of the second vector on the tip (head) of the first
vector as shown
.
The resultant of the vector addition is a vector that goes from the tail
of the first vector to the tip of the second vector,
.
Since the direction of
set the
direction of the acceleration, it can be seen that the direction of the
acceleration is toward the center of the circle.
Thus, this
acceleration is called centripetal acceleration (average
acceleration for discrete
).
Notices that independent of the initial and final position of the point in the
circle, the same derivation will lead to the previous conclusion about the
direction of the acceleration.

Following the analysis of the diagrams sequence, the magnitude of this
acceleration can also be obtained (magnitude of the centripetal acceleration).
To begin this derivation, let us focus the attention on the triangle presented in the
drawing,
.
As shown in the diagram, the angle
is related to half
of the difference between the final and initial angle by the relation
. This
angle is
additionally present in the various other parts of the diagram,
.
The remaining angles are of value
,
.
With all this information, the angles of the dark triangle are known,
.
Next, the yellow triangle
is formed by the radius extended from the initial and the final position of the
point to the center of the circle and, the third side, the arc length going from
the initial position to the final position. From the figure, two of the angles
of this triangle are of value
while the other one
is
(notice that the
blue triangle has a 90^{0} angle and an
angle implying that
the angles in the yellow triangle are also of value
). In addition,
the yellow and black triangles are equivalent to each other; therefore, the
following proportion between the sides of the triangles can be established
.
Substituting this result in the expression for the acceleration of the point,
,
the average magnitude of the centripetal acceleration becomes
.

Thus,

Average centripetal acceleration

Magnitude:

Direction: (toward the center of the circle)

The instantaneous centripetal
acceleration is obtained in the limit when
. In this case, the
angle difference .
However, the ratio
.
Remember that for small angles,
.
Therefore, the instantaneous centripetal acceleration is