5.7

Vertical Circular Motion

Motorcycle stunt drivers perform a feat in which they drive their cycles around a vertical circular track, as in Figure 5-21a. Usually, the speed varies in this stunt. When the speed of travel on a circular path changes from moment to moment, the motion is said to be “nonuniform.” Nonetheless, we can use the concepts that apply to uniform circular motion to gain considerable insight into the motion that occurs on a vertical circle.

Figure 5-21 (a) A vertical loop-the-loop motorcycle stunt. (b) The normal force and the weight of the cycle and the rider are shown here at four locations.

There are four points on a vertical circle where the centripetal force can be identified easily, as Figure 5-21b indicates. As you look at Figure 5-21b, keep in mind that the centripetal force is not a new and separate force of nature. Instead, at each point the centripetal force is the net sum of all the force components oriented along the radial direction and points toward the center of the circle. The drawing shows only the weight of the cycle plus rider (magnitude = mg) and the normal force pushing on the cycle (magnitude = FN). The propulsion and braking forces are omitted for simplicity, because they do not act in the radial direction. The magnitude of the centripetal force at each of the four points is given as follows in terms of mg and FN:

Problem solving insight Centripetal force is the name given to the net force that points toward the center of a circular path. As shown here, there may be several forces that contribute to this net force.

As the cycle goes around, the magnitude of the normal force changes. It changes because the speed changes and because the weight does not have the same effect at every point. At the bottom, the normal force and the weight oppose one another, giving a centripetal force of magnitude . At the top, the normal force and the weight reinforce each other to provide a centripetal force whose magnitude is . At points 2 and 4 on either side, only FN2 and FN4 provide the centripetal force. The weight is tangent to the circle at points 2 and 4 and has no component pointing toward the center. If the speed at each of the four places is known, along with the mass and the radius, the normal forces can be determined.
Riders who perform the loop-the-loop trick must have at least a minimum speed at the top of the circle to remain on the track. This speed can be determined by considering the centripetal force at point 3. The speed v3 in the equation is a minimum when FN3 is zero. Then, the speed is given by . At this speed, the track does not exert a normal force to keep the cycle on the circle, because the weight mg provides all the centripetal force. Under these conditions, the rider experiences an apparent weightlessness like that discussed in Section 5.6, because for an instant the rider and the cycle are falling freely toward the center of the circle.

The physics of the loop-the-loop motorcycle stunt.

Sections 5.4–5.7 discuss a number of applications of centripetal acceleration and centripetal force:

· | Banked Curves
· | Satellites in Circular Orbits
· | Apparent Weightlessness and Artificial Gravity
· | Vertical Circular Motion

Self-Assessment Test 5.2 can be used to check your understanding of the physics concepts on which these applications are based.

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.