Explanation The below figure represents the free body diagram of the slender rod with the particle attached to the rod. Figure (1) Here, initial center of mass is G 1 and final center of mass is G 2 . Write the expression for new center of gravity of the system when particle attached to the rod: x G 2 = M x 1 + m x 2 M + m (I) Here, mass of the particle is m , mass of the rod is M , distance of center of gravity from the origin is x 1 and the distance of centre of gravity of small particle m to the point of attachment on the rod is x 2 . Write the expression for the distance of m to the new center of gravity G 2 : x G = 3 L 4 − x G 2 (II) Here, the total length of the rod is L . Write the expression for “conservation of linear momentum” at x -direction: M V m = ( M + m ) v x v x = M V M M + m (III) Here, initial velocity of the rod before impact is V m , final velocity of combined system after the impact in x -direction is v x . Write the expression for “conservation of linear momentum” at y -direction: M V m = ( M + m ) v y v y = M V M M + m (IV) Here, initial velocity of the rod before impact is V m , final velocity of combined system after the impact in y -direction is v y . Write the expression for moment of inertia of entire rod system using “Parallel axis theorem”: I = 1 12 M L 2 + M ( L 4 − M L 4 M + m ) 2 + M ( M L 4 M + m ) 2 (V) Write the expression for the principle of “Conservation of angular momentum” for the system. m V m x G = I ω (VI) Conclusion: Substitute L 2 for x 1 (since the center of gravity lies at the mid-point of its length), 3 L 4 for x 2 .in the Equation (I). x G 2 = M × L 2 + m × 3 L 4 m + M Substitute M × L 2 + m × 3 L 4 m + M for x G 2 in Equation ( II ) for the distance from m to the new center of gravity

8th Edition

ISBN: 9781119047315

Author: Bolton

Publisher: VST

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Chapter 6.9, Problem 175RP

To determine

The final linear of the bar with the combined particle after impact.

The angular velocity of the bar with the combined particle after impact.

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Chapter 6.9, Problem 174RP

Chapter 6.9, Problem 176RP

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The final linear of the bar with the combined particle after impact. The angular velocity of the bar with the combined particle after impact.