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Circular Motion

With the subject of motion in more than one dimension broached, we can consider things other than simple linear motion. For example, many types of motion involve curved paths rather than linear ones. We have already seen one example in the parabolic shape of the trajectory of an object in free-fall. Another example is simply the motion of an object in a circular path. This includes motions as diverse as cars on a Ferris wheel to satellites in orbit around the earth. This kind of motion is somewhat special in that the object of interest maintains a constant speed as it goes around the circle, i.e. the magnitude of its velocity doesn't change. However, the direction of the velocity does have to change continuously to maintain the circular path. Whenever the velocity magnitude, direction, or both is changed, there must be a non-zero acceleration acting to change the motion. Look at the figure below. We've already discussed radians and their connection to radii and arc-lengths. Let's review how they would apply to finding the acceleration of an object in a circular path by looking at the figure below.

The object moves in a circular path of radius R. At some time, t_0, the velocity has direction shown in the figure and magnitude v. Some short time later, at t_1, the object has a velocity with a new direction but the same old magnitude, v. The angular position of the object has changed by an amount . The velocity change is, as shown, . The distance traveled by the object in time is the arc-length . We can now start to relate the items we have defined. First, notice that, by the definition of radians (see your lecture notes from the first week), we have

where we have only talked about magnitudes. The direction for we get by noting that it is approximately perpendicular to by our picture. Since both and are equal to , then they are equal to each other, so

The definition of average acceleration during the time is , so

where we remember that the magnitude of the objects velocity, v, and the radius R are constant in time. The time taken for the object to travel arc-length is set by the time difference and the velocity, so we know that . Therefore, our average acceleration is just

As usual, we move from the average acceleration (which is just the average rate of change of the velocity) to the exact instantaneous acceleration by taking the limit as . This is the same as letting . In that case, the change in velocity is exactly perpendicular to . But since is tangent to the circular path and therefore perpendicular to R, then the change in velocity is parallel or anti-parallel to R. From our picture, we see that is anti-parallel (it points in toward the center while the radius points outward). Therefore, the acceleration needed to maintain a perfectly circular path of radius R with constant speed is

with direction inward toward the center of the circular path. When such a condition prevails, we say that the object is subject to a centripetal acceleration and that the object is undergoing uniform circular motion.

Of course, for an acceleration to be acting, there must be a force. Any force that provides an acceleration satisfying the centripetal condition (i.e. center-pointing, magnitude constant at ) is a centripetal force. A string tied to a rock whirled at constant velocity in a horizontal circle provides a centripetal force through the tension supplied by your hand. For the Ferris wheel, the tension is provided by struts which connect the seats to the wheel and the wheel to its center. Satellites are maintained in orbit by earth's gravity.