(a) Explanation Given information: The time ( t ) required to bring the piston rod of an air cylinder to rest is 2 sec. Calculation: Show the a-t curve for the piston rod during the time ( t ) 2 sec as in Figure (1). Take the approximate value for acceleration (- a ) corresponding to time ( t ) from Figure 1. Tabulate the approximate value of acceleration (- a ) as in Table (1). time ( t ) (sec) acceleration ( -a ) ( m/s 2 ) 0 -4 0.25 -2.43 0.50 -1.40 0.75 -0.85 1 -0.50 1.25 -0.28 1.50 -0.13 1.75 -0.06 2 0 Write the expression to calculate the average acceleration ( a ave ) : ( a ave ) i = a i + a f 2 (1) Here, a i is initial acceleration particular time and a f is final acceleration of particular time. Calculate the average acceleration ( a ave ) using equation (1): ( a ave ) 1 = a 0 + a 0.25 2 Here, a 0 is acceleration at time 0 sec and a 0.25 is acceleration at time 0.25 sec. Substitute − 4 m/s 2 for a 0 and − 2.43 m/s 2 for a 0.25 . ( a ave ) 1 = − 4 + ( − 2.43 ) 2 = − 3.215 m/s 2 Similarly calculate the values of average acceleration a ave for different time values. Tabulate the calculated values of average acceleration a ave as in Table (2). time ( t )(sec) acceleration ( -a ) ( m/s 2 ) average acceleration ( a ave ) ( m/s 2 ) 0 -4 0.25 -2.43 -3.215 0.50 -1.40 -1.915 0.75 -0.85 -1.125 1 -0.50 -0.675 1.25 -0.28 -0.390 1.50 -0.13 -0.205 1.75 -0.06 -0.095 2 0 -0.030 Write the expression for calculating the change in velocity ( Δ v ) for specified interval of time ( Δ t ) using the relation: Δ v = a ave Δ t (2) Here, Δ t is change in time. Calculate the change in velocity between the time 0 sec to 0.25 sec. Substitute − 3.215 m/s 2 for a ave and 0.25 sec for Δ t in equation (2). Δ v = − 3.215 × 0.25 = − 0 (b) To determine The distance ( x 2 ) traveled by the piston rod as it is brought to rest.

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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Chapter 11.3, Problem 11.81P

(a)

To determine

The initial velocity (v0) of the piston rod.

(b)

To determine

The distance (x2) traveled by the piston rod as it is brought to rest.

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Chapter 11.3, Problem 11.80P

Chapter 11.3, Problem 11.82P

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The initial velocity ( v 0 ) of the piston rod.

Solution Summary: The author shows the a-t curve for the piston rod during the time ( t ) 2 sec as in Figure 1.