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Mobile in Uniform Circular Motion

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.

 

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Centripetal 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 900 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

Instantaneous centripetal acceleration Magnitude:
Direction: (toward the center of the circle)

 

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Example

 

  1) Question

 

a)

-19.3 KJ

N

b)

1028.8m and 0.69g

 

c)

13333.3m and 069g

 

d)

-29.5 KJ

 

e)

None of the above.

 

 

 

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