(a) Explanation Write the expression for the moment of inertia of the rod using parallel axis theorem. I z r = m r r 2 4 + m r b 2 12 = m r 12 [ 3 ( r 2 ) + b 2 ] (I) Here, the moment of inertia of the rod in its initial position is I z r , the mass of the rod is m r , the length of the rod is b and the radius of the rod is r . Write the expression for the angular momentum of the rod A . H r = I z r ω 1 (II) Here, the initial angular velocity of the rod ω 1 . Substitute m r 12 [ 3 ( r 2 ) + b 2 ] for I z r in Equation (II). H r = ( m r 12 [ 3 ( r 2 ) + b 2 ] ) ω 1 (III) Write the expression for the moment of inertia of cylinder. I z c y l = m c y l ( r i 2 + r 0 2 ) 2 + m c y l 12 ( b 2 ) 2 (IV) Here, the mass of cylinder is m c y l , inner radius of the cylinder is r i , the outer radius of the cylinder is r 0 . Substitute r for r i and 2 r for r 0 in Equation (IV). I z c y l = m c y l ( r 2 + ( 2 r ) 2 ) 2 + m c y l 12 ( b 2 ) 2 = m c y l ( r 2 + ( 4 r 2 ) ) 2 + m c y l 12 ( b 2 ) 2 = m c y l ( 5 r 2 ) 2 + m c y l 12 ( b 2 ) 2 = m c y l 12 [ 30 ( r 2 ) + ( b 2 ) 2 ] (V) Write the expression for the angular momentum of the cylinder B . H c y l = I z c y l ω 1 (VI) Substitute m c y l 12 [ 30 ( r 2 ) + ( b 2 ) 2 ] for I z c l y in Equation (VI). H c y l = ( m c y l 12 [ 30 ( r 2 ) + ( b 2 ) 2 ] ) ω 1 Write the expression for the initial angular momentum of the system. H i = H r + H c y l (VII) Substitute ( m c y l 12 [ 30 ( r 2 ) + ( b 2 ) 2 ] ) ω 1 for H c y l and ( m r 12 [ 3 ( r 2 ) + b 2 ] ) ω 1 for H r in Equation (VII). H i = ( m r 12 [ 3 ( r 2 ) + b 2 ] ) ω 1 + ( m c y l 12 [ 30 ( r 2 ) + ( b 2 ) 2 ] ) ω 1 (VIII) Draw the schematic diagram for the rod. Figure-(1) Write the expression for the moment of inertia of the rod in the final position. I z r o d 2 = I z r + m c y l d z 2 (VIII) Here, the distance between the axis of the system and the centroidal axis of the rod is d z . Substitute m r 12 [ 3 ( r 2 ) + b 2 ] for I z r and b 4 for d z in Equation (VIII). I z , r o d 2 = m r 12 [ 3 ( r 2 ) + b 2 ] + m c y l ( b 4 ) 2 (IX) Write the expression for the angular momentum of the rod. H z , r o d 2 = I z , r o d 2 ω 2 (X) Here, the final angular velocity is ω 2 . Substitute m r 12 [ 3 ( r 2 ) + b 2 ] + m c y l ( b 4 ) 2 for I z , r o d 2 in Equation (X). H z , r o d 2 = ( m r 12 [ 3 ( r 2 ) + b 2 ] + m c y l ( b 4 ) 2 ) ω 2 Write the expression for the final angular momentum of the cylinder. H c y l 2 = I c y l 2 ω 2 (XI) Here, the moment of inertia of the rod in the final position is I c y l 2 . Substitute ( m c y l 12 [ 30 ( r 2 ) + ( b 2 ) 2 ] ) ω 1 for I c y l 2 in Equation (XI). H c y l 2 = ( m c y l 12 [ 30 ( r 2 ) + ( b 2 ) 2 ] ) ω 2 Write the expression for the final angular momentum of the system (b) To determine The final angular velocity of the system, just before the rod leave the cylinder. (c) To determine The final angular velocity of the system, just after the rod leave the cylinder.

8th Edition

ISBN: 9781118885840

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

Publisher: WILEY

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Chapter 6.9, Problem 190RP

(a)

To determine

The angular velocity of the system, if the inner rod A has moved so that a length b/2 is protruding from the cylinder.

(b)

To determine

The final angular velocity of the system, just before the rod leave the cylinder.

(c)

To determine

The final angular velocity of the system, just after the rod leave the cylinder.

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Chapter 6.9, Problem 189RP

Chapter 6.9, Problem 191RP

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The angular velocity of the system, if the inner rod A has moved so that a length b / 2 is protruding from the cylinder.

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The angular velocity of the system, if the inner rod A has moved so that a length b/2 is protruding from the cylinder.

The final angular velocity of the system, just before the rod leave the cylinder.

The final angular velocity of the system, just after the rod leave the cylinder.

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