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Kinematics is the study of the geometry of motion. Of the most common of moving objects, the rolling wheel is probably the most studied. This presentation shows how the velocity of particular points on a rolling wheel are determined using the relative motion equation. The most important aspect of the relative motion equation is that it is a vector equation, though it is usually applied only for plane 2D motion. In a later presentation, the acceleration of particular points on a rolling wheel will be developed.
As shown in the presentation, plane motion can be separated into a combination of “pure” translation plus “pure” rotation. For a rolling wheel, the velocity of pure translation is the velocity of the center of the wheel. The velocity of pure rotation is the radius of the wheel times the angular velocity of the wheel and is directed tangent to the wheel at that point. Note that due to the “no slip” condition at the point of contact, where the velocity is zero, results in the velocity of the center of the wheel to also be the radius of the wheel times the angular velocity of the wheel. An interesting result is that the velocity of the point at the top of the wheel is twice the velocity of the center of the wheel.
Also, note that the curve of a point on the wheel as it makes one complete revolution is called a “cycloid” and has very interesting mathematical properties.
The relative motion equations will be applied to several more dynamic systems. Understanding the kinematics of the rolling wheel will make understanding these mechanisms considerably easier.
-Tom
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