Explanation Given information: The length of chain is L . Calculation: Consider the mass of the chain between the two piles is m , the time period is between t and t + Δ t , and Δ m be the amount of chain that is picked up at A and deposited at C . At time t , Δ m is in pile A and it has the downward speed v just above the pile C . Hence, the remaining , mass ( m − Δ m ) is moving with speed v . At time t + Δ t , Δ m is moving with the speed v + Δ v just above the pile A and it is placed in pile C . For over the time period an unbalanced weight of chain acts on the system. Consider the mass per unit length of chain as m L = ρ . Assume the reaction from the floor is equal to the weight of the chain in contact with the floor. Calculate the weight of chain ( W ) as shown below. W = ρ g h Substitute m l for ρ . W = m L g h The rate of mass of the chain Δ m = m L v ( Δ t ) . Sketch the Free Body Diagram of the chain as shown in Figure 1. Refer to Figure 1. Apply the impulse and momentum principle as shown below. r [ ( Δ m ) v + ( m − Δ m ) v ] + r W ( Δ t ) = r [ ( Δ m ) ( v + Δ v ) + ( m − Δ m ) ( v + Δ v ) ] Divide the equation by r and remove cancel the terms ( Δ m ) v and ( m − Δ m ) v as shown below. 0 + W ( Δ t ) = ( Δ m ) ( v + Δ v ) + ( m − Δ m ) ( Δ v ) W Δ t = Δ m v + Δ m Δ v + m Δ v − Δ m Δ v W Δ t = Δ m v + m Δ v Substitute m L g h for W and m L v ( Δ t ) for Δ m

Vector Mechanics for Engineers: Statics and Dynamics

11th Edition

ISBN: 9780073398242

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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Chapter 14.3, Problem 14.88P

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The speed v of the chain at time t.

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Chapter 14.3, Problem 14.87P

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The speed v of the chain at time t .

Solution Summary: The author calculates the mass per unit length of chain as mL=rho .

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The speed v of the chain at time t.

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Chapter 14 Solutions