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An object moves in a circle at constant speed. Although its speed is constant, its velocity is not. Velocity is a vector, with magnitude and direction. In order to move along the circular path its direction must change, so it accelerates. The direction of the acceleration is easy to figure: It cannot be along the direction of v, because if it were the magnitude of v would change and, as stated, the object has constant speed. Thus the acceleration must be perpendicular to v, and as it clearly is not outward from the center it must be inward. That's why we call it a centripetal, or center seeking, acceleration.
The magnitude of the acceleration is another matter. There are several ways to calculate it - I show one below. To follow the work you will need to use a fair amount of mathematics.
Look at the picture: It is a circle of radius r. An object at point P is moving around the circle at constant speed. Its velocity is tangent to the circle at P.
and in terms of the angle 2
But sin 2 is yp/r and cos 2 = xp/r so we can write
Since acceleration is the rate of change of velocity, and we can write the x- and y- components in terms of sin(2) and cos(2), we have
The magnitude of a is just the square root of a . a, which gives us the desired result,
a = v2/r.