You probably recall that the change in velocity with time is equal to the acceleration. Since velocity is a vector, an object is accelerated when either the magnitude or the direction of its velocity changes. This is a new concept since until now we have considered only acceleration that results from changes in the magnitude of the velocity. As previously mentioned, the velocity of an object in uniform circular motion is constantly changing direction. Thus, this motion involves acceleration. This is called centripetal acceleration, and is denoted by ac. Let's consider what happens when our object moves a small distance along the circle. The change in the velocity vector, Dv, is represented by vector v2 minus vector v1. We can graphically subtract the vector v1 from vector v2 to get vector Dv. Notice that Dv points towards the center of the circle. In uniform circular motion, the Dv vector always points towards the center of the circle it is describing, regardless of the object's position on the circle. We know the direction of Dv, which indicates the direction of the centripetal acceleration. Now let's find an expression for Dv/Dt to get the magnitude of the centripetal acceleration. If you look closely at the triangle formed by Dv equal to v2 minus v1, q is the apex angle of the triangle. Since the magnitude of the velocity does not change, the magnitudes of v2 and v1 also remain the same. Thus v equals v1 equals v2. Since the triangle has two sides that are the same, it is an isosceles triangle. Now notice that the distance traveled by the object along the arc length is equal to the velocity times Dt. We can see that the small triangle formed by the radius and arc length is also an isosceles triangle with the angle of displacement equal to q. Since the two triangles are similar triangles, the ratios of two sides of each triangle are equal to each other. Lastly, we replace the arc length by vDt. From this expression we solve for Dv over Dt, which is the centripetal acceleration. We find that the centripetal acceleration is equal to velocity squared divided by the radius. We found earlier that the direction of the centripetal acceleration always points towards the center of the circle. Thus we have fully defined the direction and magnitude of the centripetal acceleration vector. The centripetal acceleration can also be written in terms of the angular velocity by using the relationship for uniform circular motion that velocity equals angular velocity times radius.

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