Ch13_Summary - bee29400_ch13_754-853.indd Page 843 4:42:18...

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843 REVIEW AND SUMMARY This chapter was devoted to the method of work and energy and to the method of impulse and momentum. In the first half of the chapter we studied the method of work and energy and its applica- tion to the analysis of the motion of particles. We first considered a force F acting on a particle A and defined the work of F corresponding to the small displacement d r [Sec. 13.2] as the quantity dU 5 F ? d r (13.1) or, recalling the definition of the scalar product of two vectors, dU 5 F ds cos a (13.1 9 ) where a is the angle between F and d r (Fig. 13.29). The work of F during a finite displacement from A 1 to A 2 , denoted by U 1 y 2 , was obtained by integrating Eq. (13.1) along the path described by the particle: U 1 y 2 5 # A 2 A 1   F ? d r (13.2) For a force defined by its rectangular components, we wrote U 1 y 2 5 # A 2 A 1  ( F x dx 1 F y dy 1 F z dz ) (13.2 0 ) The work of the weight W of a body as its center of gravity moves from the elevation y 1 to y 2 (Fig. 13.30) was obtained by substituting F x 5 F z 5 0 and F y 5 2 W into Eq. (13.2 0 ) and integrating. We found U 1 y 2 5 2 # y 2 y 1   W dy 5 Wy 1 2 Wy 2 (13.4) Work of a force Work of a weight A 1 s 1 s 2 s A 2 F O A d r d s a Fig. 13.29 A 2 A A 1 y 2 y 1 dy y W Fig. 13.30
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844 Kinetics of Particles: Energy and Momentum Methods The work of a force F exerted by a spring on a body A during a finite displacement of the body (Fig. 13.31) from A 1 ( x 5 x 1 ) to A 2 ( x 5 x 2 ) was obtained by writing dU 5 2 F dx 5 2 kx dx U 1 y 2 5 2 # x 2 x 1   kx dx 5 1 2 kx 2 1 2 1 2 kx 2 2 (13.6) The work of F is therefore positive when the spring is returning to its undeformed position . Work of the force exerted by a spring A 0 A 1 Spring undeformed B B B F A A 2 x 1 x x 2 Fig. 13.31 Fig. 13.32 O A 2 A 1 r 2 r 1 q d r F F M r A ' A m d q The work of the gravitational force F exerted by a particle of mass M located at O on a particle of mass m as the latter moves from A 1 to A 2 (Fig. 13.32) was obtained by recalling from Sec. 12.10 the expres- sion for the magnitude of F and writing U 1 y 2 5 2 # r 2 r 1   GM m r 2 dr 5 GM m r 2 2 GM m r 1 (13.7) The kinetic energy of a particle of mass m moving with a velocity v [Sec. 13.3] was defined as the scalar quantity T 5 1 2 mv 2 (13.9) Work of the gravitational force Kinetic energy of a particle
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845 From Newton’s second law we derived the principle of work and energy, which states that the kinetic energy of a particle at A 2 can be obtained by adding to its kinetic energy at A 1
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