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

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

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Chapter 6 Circular Motion and Other Applications of Newton’s Laws CHAPTER OUTLINE 6.1 Newton’s Second Law Applied to Uniform Circular Motion 6.2 Nonuniform Circular Motion 6.3 Motion in Accelerated Frames 6.4 Motion in the Presence of Resistive Forces 6.5 Numerical Modeling in Particle Dynamics ± The London Eye, a ride on the River Thames in downtown London. Riders travel in a large vertical circle for a breathtaking view of the city. In this chapter, we will study the forces involved in circular motion. (© Paul Hardy/CORBIS) 150

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I 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 compli- cated. For example, we shall apply Newton’s laws to objects traveling in circular paths. Also, we shall discuss motion observed from an accelerating frame of reference and motion of an object through a viscous medium. For the most part, this chapter consists of a series of examples selected to illustrate the application of Newton’s laws to a wide variety of circumstances. 6.1 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 that has a magnitude The acceleration is called centripetal acceleration because a c is directed toward the center of the circle. Furthermore, a c 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 frictionless table. Why does the ball move in a circle? According to Newton’s ﬁrst law, the ball tends to move in a straight line; however, the string prevents a c ± v 2 r 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 circle keeps the ball moving in its circular path. An athlete in the process of throwing the hammer at the 1996 Olympic Games in Atlanta, Georgia. The force exerted by the chain causes the centripetal acceleration of the hammer. Only when the athlete releases the hammer will it move along a straight-line path tangent to the circle. Mike Powell / Allsport / Getty Images 151
152 CHAPTER 6 Circular Motion and Other Applications of Newton’s Laws motion along a straight line by exerting on the ball a radial force F r that makes it fol- low the circular path. This force is directed along the string toward the center of the circle, as shown in Figure 6.1. If we apply Newton s second law along the radial direction, we nd that the net force causing the centripetal acceleration can be evaluated: (6.1) A force causing a centripetal acceleration acts toward the center of the circular path and causes a change in the direction of the velocity vector. If that force should van-

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

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