Explanation Given: The mass of the block is 20 kg . Initially the block is at rest. The stiffness of the spring A is k A = 500 N/m . The stiffness of the spring B is k B = 800 N/m . The distance between the bumpers A and B is 0.5 m . Let, s A be the deformation of the bumper A and s B be the deformation of the bumper B . The displacement of both the springs are negative because they are below datum. Calculate total deformation of springs due to the block. s = − ( s A + s B ) − 0.5 m Write the formula for kinetic energy ( T ) . T = 1 2 m v 2 (I) Here, m is the mass and v is the velocity. Refer Figure (1). Write the formula for elastic potential energy ( V e ) . V e = 1 2 k s 2 (II) Here, k is the stiffness of the spring and s is the displacement. Write the formula for gravitational potential energy ( V g ) . V g = m g s (III) Here, m is the mass of the block, g is the acceleration due to gravity and s is the displacement. Write the formula total potential energy ( V ) . V = V g + V e (IV) Here, V g is gravitational potential energy and V e is elastic potential energy. Write the formula for conservation of energy. T 1 + V 1 = T 2 + V 2 (V) Here, T 1 , T 2 are the initial and final kinetic energy and V 1 , V 2 are the initial and final potential energy. Conclusion: At Initial: Calculate the initial kinetic energy and potential energy of the block. When, the block is at rest. The kinetic energy of the block is T 1 = 0 . The potential energy of the block is V 1 = 0 . At Final: Calculate the final kinetic energy. When, the block becomes rest. T 2 = 0 Calculate the elastic potential energy ( V e ) A of the spring A . Substitute 500 N/m for k and s A for s in Equation (II). ( V e ) A = 1 2 ( 500 N/m ) s A 2 = 250s A 2 Calculate the elastic potential energy ( V e ) B of the spring B . Substitute 500 N/m for k and s A for s in Equation (II). ( V e ) B = 1 2 ( 800 N/m ) s B 2 = 400s B 2 Calculate the gravitational potential energy. Substitute 20 kg for m , 9.81 m/s 2 for g and − ( s A + s B ) − 0.5 m for s in Equation (III). V g = ( 20 kg ) ( 9.81 m/s 2 ) [ − ( s A + s B ) − 0.5 m ] = 196.2 [ − ( s A + s B ) − 0.5 ] = − 196.2 ( s A + s B ) − 98.1 Calculate the total potential energy ( V ) . Substitute V 2 for V , 250 s A 2 for ( V e ) A , 400 s B 2 for ( V e ) B and − 196

Engineering Mechanics: Statics & Dynamics (14th Edition)

14th Edition

ISBN: 9780133915426

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 14.5, Problem 87P

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The maximum deformation of the springs A and B due to collision.

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Chapter 14.5, Problem 86P

Chapter 14.5, Problem 88P

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Engineering Mechanics: Statics & Dynamics (14th Edition)

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The maximum deformation of the springs A and B due to collision.

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