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A while the package travels along the semicircular

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Unformatted text preview: table. (a) While the package travels along the semicircular track, the only forces acting are gravity and the reaction force from the track. The reaction force is directed radially. The motion is along the arc of a circle of radius R so that the reaction force does no work. Since gravity is a conservative force, total energy is conserved. Thus, the sum of the kinetic and potential energy is constant, wherefore 1 1 mv 2 + mg · 0 = mv 2 + mgh 2o 2 =⇒ 2 vo = v 2 + 2gh Also, at the moment the package reaches the upper surface, we are given that the normal force is zero. Thus, since gravity is the only nonzero force acting on the package at the moment it reaches the upper surface, we have v2 ma = m = mg =⇒ v 2 = gR R Combining the equations above, there follows 2 vo = gR + 2gh = g (R + 2h) Therefore, the velocities v and vo are v= gR and vo = g (R + 2h) 4.12. CHAPTER 4, PROBLEM 12 281 (b) Because friction is a non-conservative force, we use the Principle of Work and Energy to determine how far the package will travel along the surface before coming to rest. Because friction opposes the motion, the work done by the friction force in sliding a distance d is U1−2 = −µmgd. So, the Work/Energy Principle tells us that 1 1 =⇒ µmgd = mv 2 U1−2 = T2 − T1 = 0 − mv 2 2 2 which tells us that v2 d= 2µg Finally, substituting for v from Part (a), we conclude that d= gR R = 2µg 2µ...
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