To determine Determine the maximum counterweight to avoid overturning of the coal car about its rear wheels. Explanation Show the free body diagram and its equivalent kinetic diagram of the coal car as in Figure (1). Take moments about point B considering counterclockwise moments as positive. ∑ M B = − m a ( 3 ) N A ( 2 + 2 ) + W c a r sin 30 ° ( 3 ) − W c a r cos 30 ° ( 2 ) − T ( 1 ) = − ( W c a r / g ) ( a ) ( 3 ) (I) Here, the normal reaction force at front wheels A is N A , weight of the coal car is W c a r , tension in the cable at C is T , mass of the coal car is m , acceleration due to gravity is g and the acceleration of the coal car is a . Apply Newton’s second law along x -axis which is the direction of motion to find the acceleration of the coal car. ∑ F x = m a T − W c a r sin 30 ° = W c a r g a (II) Show the free body diagram of the length of the cables as in Figure (2). Consider the length of the cable carrying the counterweight. L = ( h − y C ) + ( y w − y C ) (III) Differentiate Equation (III) with respect to time on both sides of the equation. d L d t = d d t ( h − y C ) + d d t ( y w − y C ) 0 = [ 0 − d y C d t ] + d y w d t − d y C d t 0 = ( 0 − v C ) + ( v w − v C ) v w = 2 v C (IV) Differentiate Equation (IV) with respect to time on both sides of the equation. d v w d t = 2 d v C d t a w = 2 a C Show the free body diagram and its equivalent kinetic diagram of pulley 2 as in Figure (3). Apply Newton’s second law along y -axis . ∑ F y = m 2 a C 2 T ' − T = m 2 a C (V) Here, tension in the cable connected to counterweight is T ' . Show the free body diagram and its equivalent kinetic diagram of counter weight as in Figure (4). Apply Newton’s second law along y -axis

8th Edition

ISBN: 9781118885840

Author: James L. Meriam, L. G. Kraige, J. N. Bolton

Publisher: WILEY

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Chapter 6.4, Problem 29P

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Determine the maximum counterweight to avoid overturning of the coal car about its rear wheels.

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