Explanation Given info: The mass of the block is m , the angle of inclination of the plane is θ , the force constant of the force is k , and the distance between the block and the spring is d . The free body diagram for the given case is as shown in figure below. Figure (1) The component m g sin θ is the parallel component down the incline accelerates the block to a distance d . The block encounters a spring and the spring force opposes the weight force and slows the block till distance x and after distance ( d + x ) the block comes momentarily at rest. The gravitational force acts on the body for the distance ( d + x ) thus, the work done by the gravitational force is, W g = m g sin θ ( d + x ) (1) Here, m is the mass of the block. g is the acceleration due to gravity. d is the distance between the block and the spring. x is the length the block compresses the string. θ is the angle of inclination. The work done by the spring force is, W s = 1 2 k ( x i − x f ) 2 (2) Here, k is the force constant of the spring. x i is the initial compression of the spring. x f is the final compression in the spring. The change in the kinetic energy is, K f − K i = 1 2 m v f 2 − 1 2 m v i 2 (3) Here, K f is the final kinetic energy. K i is the initial kinetic energy. v f is the final velocity. v i is the initial velocity. From the work energy theorem the work by all conservative forces is equal to the change in the kinetic energy. W g + W s = K f − K i Substitute m g sin θ ( d + x ) for W g , 1 2 k ( x i − x f ) 2 for W s , 1 2 m v f 2 − 1 2 m v i 2 for K f − K i in the above equation. m g sin θ ( d + x ) + 1 2 k ( x i ) 2 − 1 2 k ( x f ) 2 = 1 2 m v f 2 − 1 2 m v i 2 The initial compression in the spring is negligible and the final velocity of the block is zero thus, substitute 0 for v f and 0 for x i in the above equation. m g sin θ ( d + x ) − 1 2 k ( x f ) 2 = 1 2 m ( 0 ) − 1 2 m ( v i 2 ) m g sin θ ( d + x ) − 1 2 k ( x f ) 2 = ( − 1 2 m ( v i 2 ) ) Divide the above equation by m both sides

Physics for Scientists and Engineers with Modern Physics

10th Edition

ISBN: 9781337671729

Author: SERWAY

Publisher: Cengage

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Chapter 7, Problem 48AP

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The distance up to the spring is compressed at the time for the block is momentarily at rest.

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Chapter 7, Problem 47AP

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Physics for Scientists and Engineers with Modern Physics

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The distance up to the spring is compressed at the time for the block is momentarily at rest.

Solution Summary: The free body diagram for the given case is as shown in figure below.

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Chapter 7 Solutions