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Illustration 5.2: Uniform Circular Motion: Fc and ac

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Uniform circular motion is an interesting mixture of one- and two-dimensional concepts. During uniform circular motion, the speed of the object must be constant. This is the uniform in uniform circular motion. So is an object moving in a circle with a constant speed accelerating?  Yes!  Why? The velocity is changing with time. Watch the animation (position is shown in meters and time is shown in seconds). The animation depicts an object moving in a circle at a constant speed. To determine the acceleration we need to consider the change in velocity for a change in time. Restart.

Since the speed does not change in time, what does change in time? It is the direction that changes with time. Draw two velocity vectors to convince yourself that the direction of the velocity changes with time. Recall that velocity has a direction (which always points tangent to the path, the so-called tangential direction) and a magnitude, and either or both can change with time. In what direction does the change in velocity point? Calculate the acceleration. It points toward the center of the circle. Since the object is accelerating, this motion must be due to a force (or a set of forces, a net force) that points solely toward the center of the circle. (Note: if the motion is nonuniform circular motion, the net force can point in another direction.) This direction—toward the center of the circle—is called the centripetal or center-seeking direction. It is often also called the radial direction, since the radius points from the center of the circle out to the object (the net force points in the opposite direction).

Therefore, for uniform circular motion, the acceleration always points toward the center of the circle. This is despite the fact that the velocity and the acceleration point in changing directions as time goes on. However, we get around this apparent difficulty in describing direction by defining the centripetal or radial direction and the tangential direction (the direction tangent to the circle). These directions change, but the velocity is always tangent to the circle, and the net force is always pointing toward the center of the circle. The following animation shows velocity and acceleration as the object undergoes uniform circular motion.

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