problem10_69

University Physics with Modern Physics with Mastering Physics (11th Edition)

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10.69: a) The distance the marble has fallen is . 2 ) 2 ( R r h r R h y - + = - - = The radius of the path of the center of mass of the marble is , r R - so the condition that the ball stay on the track is ). ( 2 r R g v - = The speed is determined from the work-energy theorem, . ) 2 1 ( ) 2 1 ( 2 2 ϖ I mv mgy + = At this point, it is crucial to know that even for the curved track, ; r v = ϖ this may be seen by considering the time T to move around the circle of radius r R - at constant speed V is obtained from , ) ( 2 Vt r R = - π during which time the marble rotates by an angle
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Unformatted text preview: T r R =-from which . r V = The work-energy theorem then states , ) 10 7 ( 2 mv mgy = and combining, canceling the factors of m and g leads to , 2 ) )( 10 7 ( R r h r R-+ =-and solving for h gives . ) 10 17 ( ) 10 27 ( r R h-= b) In the absence of friction, , ) 2 1 ( 2 mv mgy = and substitution of the expressions for y and 2 v in terms of the other parameters gives , 2 ) )( 2 1 ( R r h r R--=-which is solved for . ) 2 3 ( ) 2 5 ( r R h-=...
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