In the sequence of drawings shown on the diagram,
point P moves in uniform circular motion in a circle of radius
. That is,
even when the direction of the velocity of point P changes, the
magnitude of its velocity is constant.
The first drawing on the left represents the vector
velocity when the point P is at an angle
to the horizontal axis, .
Since the direction of the vector velocity is changing at all locations,
it is convenient to express the vector velocity in terms of an
orthogonal coordinate system rotating with point P rather than
the traditional xy-coordinate system, .
This coordinate system is defined in base of the unit vectors
. As it can
be seeing in the diagram, the vector velocity has only component along
where is the
magnitude of the velocity.
The acceleration of point
P can be obtained from the derivative of the velocity with
respect to the time,
In this case, the unit vector
constant with time.
At this point, it is convenient to find the functional dependences of
in terms of
In this diagram the lighter vectors represent those components. Thus,
the value of those components can be obtained in terms of the angle
. Notice that
several parts of the diagram,
Finally, the unit vectors are
because the unit vectors
and are time
independent. Finally, this derivative is
Therefore, the acceleration is
The direction of this acceleration is toward the center of the circle ().
Moreover, the angle
, the radius
of the circle
, and the arc
related by ,
From this relation, the derivative of the angle with respect to the time
can be calculated,
The derivative of the arc length with respect to the time is precisely
the definition of the magnitude of the velocity (remember that the
magnitude of the instantaneous velocity is the instantaneous speed).
and the acceleration is
This result is the same as the result obtained for the centripetal
acceleration when derived without the explicit use of calculus.