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p0404

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Unformatted text preview: 198 CHAPTER 4. WORK AND ENERGY 4.4 Chapter 4, Problem 4 Problem: A ball of mass m rests against the bearing plate of mass mp in a child’s spring gun. The bearing plate is attached to a spring of constant k that is initially compressed through a distance ξ . The spring is unstretched when x = 0. Friction effects are negligible. The speed of the ball as it exits the gun, V , is 80% of the value it would achieve if the bearing plate’s mass were negligible. Determine the mass of the bearing plate. ......... ................................................................................................................................................................................................................... . . ..... . . . . . . . . . . . ........... . . . . . . .. . . . . .. . . . . .... . .. .......... .... ..... ...... ... ... ... .. . ..... .. .. . . . . . . ......................... . . . . . . .... . . . . .............................................................................................................................................. .............................................................................................................................................................................................. . 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. . . . . . . ............... . ................. . ................. ................ . . .. .. .. .. . • k • m v ξ x Solution: We can solve using the Principle of Work and Energy. Since the only force acting is the spring force, F = −kx, we have 0 U1−2 = − −ξ kx dx = − 12 kx 2 x=0 = x=−ξ 12 kξ 2 Before the spring is released, the kinetic energy is zero. When the ball reaches x = 0, the kinetic energy is the combined energy of both the bearing plate and the ball. Thus, T1 = 0, T2 = 1 (m + mp ) V 2 2 The Principle of Work and Energy tells us that U1−2 = T2 − T1 , wherefore 1 1 (m + mp ) V 2 = k ξ 2 2 2 Solving for V yields V =ξ k m + mp If the mass of the bearing plate were negligible, this equation would simplify to Vo = ξ We are given V = 4 Vo , which tells us that 5 m 4 = m + mp 5 Solving for mp , we conclude that mp = 9 m 16 =⇒ m= 16 (m + mp ) 25 k m ...
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