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Derivation of the Centripetal Acceleration

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 with respect 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 and . As it can be seeing in the diagram, the vector velocity has only component along the component, 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 is not constant with time.

At this point, it is convenient to find the functional dependences of and in terms of the standard and unit vectors, . 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 the angle appears in several parts of the diagram, . Finally, the unit vectors are

Thus, because the unit vectors and are time independent. Finally, this derivative is   which 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 length are 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). Thus, 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.

 

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by Luis F. Sez, Ph. D.    Comments and Suggestions: LSaez@dallaswinwin.com