problem05_p125

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

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5.125: Take all accelerations to be positive downward. The equations of motion are straightforward, but the kinematic relations between the accelerations, and the resultant algebra, are not immediately obvious. If the acceleration of pulley B is , B a then , 3 a a B - = and B a is the average of the accelerations of masses 1 and 2, or . 2 2 3 2 1 a a a a B - = = + There can be no net force on the massless pulley B , so . 2 A C T T = The five equations to be solved are then . 0 2 0 2 3 2 1 3 3 3 2 2 2 1 1 1 = - = + + = - = - = - C A C A A T T a a a a m T g m a m T g m a m T g m These are five equations in five unknowns, and may be solved by standard means. A symbolic-manipulation program is of great use here. a) The accelerations 1 a and 2 a may be eliminated by using ))). 1 ( ) 1 (( 2 ( ) ( 2 2 1 2 1 3 m m T g a a a A + - - = + - = The tension A T may be eliminated by using ). ( ) 2 1 ( ) 2 1 ( 3 3 a g m T T C A - = = Combining and solving for 3 a gives . 4 4 3 1 3 2 2 1 3 1 3 2 2 1 3 m m m m m m m m m m m m g a + + + + - = b) The acceleration of the pulley B has the same magnitude as 3 a and is in the opposite direction. c) ). ( 2 2 3 1 3 1 1 1 a
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Unformatted text preview: g m m g m T g m T g a C A--=-=-= Substituting the above expression for 3 a gives . 4 3 4 3 1 3 2 2 1 3 1 3 2 2 1 1 m m m m m m m m m m m m g a + + +-= d) A similar analysis (or, interchanging the labels 1 and 2) gives . 4 3 4 3 1 3 2 2 1 3 2 3 1 2 1 2 m m m m m m m m m m m m g a + + +-= e) & f) Once the accelerations are known, the tensions may be found by substitution into the appropriate equation of motion, giving . 4 8 T , 4 4 3 1 3 2 2 1 3 2 1 C 3 1 3 2 2 1 3 2 1 m m m m m m m m m g m m m m m m m m m g T A + + = + + = g) If m m m = = 2 1 and , 2 3 m m = all of the accelerations are zero, mg T C 2 = and . mg T A = All masses and pulleys are in equilibrium, and the tensions are equal to the weights they support, which is what is expected....
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This document was uploaded on 02/04/2008.

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