6 - Circular Motion and Other Applications of Newton's Laws

6 - Circular Motion and Other Applications of Newton's Laws...

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c h a p t e r Circular Motion and Other Applications of Newton’s Laws This sky diver is falling at more than 50 m/s (120 mi/h), but once her para- chute opens, her downward velocity will be greatly reduced. Why does she slow down rapidly when her chute opens, en- abling her to fall safely to the ground? If the chute does not function properly, the sky diver will almost certainly be seri- ously injured. What force exerted on her limits her maximum speed? (Guy Savage/Photo Researchers, Inc.) 6.1 Newton’s Second Law Applied to Uniform Circular Motion 6.2 Nonuniform Circular Motion 6.3 (Optional) Motion in Accelerated Frames 6.4 (Optional) Motion in the Presence of Resistive Forces 6.5 (Optional) Numerical Modeling in Particle Dynamics C h a p t e r O u t l i n e 151 P U Z Z L E R P U Z Z L E R
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n the preceding chapter we introduced Newton’s laws of motion and applied them to situations involving linear motion. Now we discuss motion that is slightly more complicated. For example, we shall apply Newton’s laws to objects traveling in circular paths. Also, we shall discuss motion observed from an acceler- ating frame of reference and motion in a viscous medium. For the most part, this chapter is a series of examples selected to illustrate the application of Newton’s laws to a wide variety of circumstances. NEWTON’S SECOND LAW APPLIED TO UNIFORM CIRCULAR MOTION In Section 4.4 we found that a particle moving with uniform speed v in a circular path of radius r experiences an acceleration a r that has a magnitude The acceleration is called the centripetal acceleration because a r is directed toward the center of the circle. Furthermore, a r is always perpendicular to v . (If there were a component of acceleration parallel to v , the particle’s speed would be changing.) Consider a ball of mass m that is tied to a string of length r and is being whirled at constant speed in a horizontal circular path, as illustrated in Figure 6.1. Its weight is supported by a low-friction table. Why does the ball move in a circle? Because of its inertia, the tendency of the ball is to move in a straight line; how- ever, the string prevents motion along a straight line by exerting on the ball a force that makes it follow the circular path. This force is directed along the string toward the center of the circle, as shown in Figure 6.1. This force can be any one of our familiar forces causing an object to follow a circular path. If we apply Newton’s second law along the radial direction, we find that the value of the net force causing the centripetal acceleration can be evaluated: (6.1) F r ma r m v 2 r a r v 2 r 6.1 152 C H A P T E R 6 Circular Motion and Other Applications of Newton’s Laws Force causing centripetal acceleration I 4.7 m F r F r r Figure 6.1 Overhead view of a ball moving in a circular path in a horizontal plane. A force F r directed toward the center of the cir- cle keeps the ball moving in its circular path.
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6.1 Newton’s Second Law Applied to Uniform Circular Motion 153
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