Explanation Given Information: The radius ( r ) of the pulley is 75 mm. The stiffness ( k ) of the spring is 2,000 N/m. The combined mass moment of inertia ( I O ) of all rotating components about point O is 0.40 kg ⋅ m 2 . The mass ( m B ) of the baseball is 145 g. The velocity ( v b ) of the baseball is 10 m/s. The distance ( h ) between point O to baseball is 0.7 m. The coefficient of restitution ( e ) between the bat and ball is 0.59. The angle ( θ ) between position 1 and 2 is 120 ° . Calculation: Initially the bat is at rest. The kinetic energy ( T 1 ) at initial position, potential energy ( V g 1 ) due to gravity at initial position, work done ( U ′ 1 − 2 ) at initial position, potential energy ( V g 2 ) due to gravity at final position, and potential energy ( V e 2 ) due to elasticity at final position are zero. Find the initial change in length of curvature ( s 1 ) using the equation. s 1 = r θ Substitute 75 mm for r and 120 ° for s 1 . s 1 = ( 75 mm × 1 m 1 , 000 mm ) ( 120 ° × π rad 180 ° ) = 0.05 π m Write the potential energy ( V e 1 ) due to elasticity at initial position. V e 1 = 1 2 k s 1 2 Substitute 2,000 N/m for k and 0.05 π m for s 1 . V e 1 = 1 2 ( 2 , 000 N/m ) ( 0.05 π ) = 24.674 J Write the equation of kinetic energy ( T 1 ) at final position. T 2 = − 1 2 I O ω 2 2 Here, ω 2 is the angular velocity of bat at impact. Substitute 0.40 kg ⋅ m 2 for I O . T 2 = − 1 2 ( 0.40 ) ω 2 2 = 0.20 ω 2 2 Consider the conservation of work energy equation. Find the angular velocity ( ω 2 ) of the Bat at impact. T 1 + V g 1 + V e 1 + U ′ 1 − 2 = T 2 + V g 2 + V e 2 Substitute 0 for T 1 , 0 for V g 1 , 24.674 J for V e 1 , 0 for U ′ 1 − 2 , 0.20 ω 2 2 for T 2 , 0 for V g 2 , and 0 for V e 2 . T 1 + V g 1 + V e 1 + U ′ 1 − 2 = T 2 + V g 1 + V e 1 0 + 0 + 24.674 + 0 = 0.20 ω 2 2 + 0 + 0 ω 2 2 = 24

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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When a bar of the uniform cross-section is acted by a load, such that the load acts along the longitudinal axis of the bar, such kinds of loadings are known as axial loadings. In general terms, a load is an external force acting on a component. An axial …

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Chapter 17, Problem 17.141RP

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The velocity of the baseball immediately after the impact.

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Chapter 17, Problem 17.140RP

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