Explanation Given: A mass of weight 4 pounds is attached to a spring and is released from a point 1 foot above the equilibrium position. The value of spring constant is 2 lb/ft . Calculation: Convert the given mass in slug as follows: m = 4 32 = 1 8 slug Therefore, the mass is 1 8 slug . A damping force is proportional to the instantaneous velocity, it means that β is 1. The general equation of motion is, m x ″ + β x ′ + k x = 0 (1) Substitute m = 1 8 , k = 2 and β = 1 in the equation (1) as, 1 8 x ″ + x ′ + 2 x = 0 The auxiliary equation of above differential equation is, 1 8 m 2 + m + 2 = 0 8 8 m 2 + 8 m + 2 ⋅ 8 = 0 m 2 + 8 m + 16 = 0 Solve the above quadratic equation as, m 2 + 8 m + 16 = 0 ( m + 4 ) 2 = 0 m = − 4 Therefore, the value of m is − 4 . The differential equation with repeated roots is expressed as, x c = c 1 e m t + c 2 t e m t (2) Substitute the value of m = − 4 in the equation (2), x c = c 1 e − 4 t + c 2 t e − 4 t (3) The mass is released from 1 foot above the equilibrium position, so x ( 0 ) is equal to − 1 and substitute x ( 0 ) = − 1 in the equation (3) as follows. x ( 0 ) = c 1 e − 4 ⋅ 0 + c 2 ⋅ 0 ⋅ e − 4 ⋅ 0 − 1 = c 1 Differentiate equation (3) as, x ′ = − 4 c 1 e − 4 t + c 2 e − 4 t − 4 c 2 t e − 4 t (4) The mass gives downward velocity of 8 ft/s so x ′ ( 0 ) is equal to 8 and substitute x ′ ( 0 ) = 8 in the equation (4) as follows, x ′ ( 0 ) = − 4 c 1 e − 4 ⋅ 0 + c 2 e

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 3.8, Problem 25E

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To find: the time at which the mass attains its extreme displacement from the equilibrium position and the position of the mass at this instant.

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Chapter 3.8, Problem 24E

Chapter 3.8, Problem 26E

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A mass weighing 20 pounds stretches a spring 6 inches. The mass is initially released from rest from a point 8 inches below the equilibrium position. (a) Find the position x of the mass at the times t = ?/12, ?/8, ?/6, ?/4, and 9?/32 s. (Use g = 32 ft/s2 for the acceleration due to gravity.)

A mass weighing 4 pounds is attached to a spring whose constant is 2 Ib/ft. The medium offers a damping force that is numerically equal to the instantaneous velocity. The mass is initially released from a point 1 foot above the equilibrium position with a downward velocity of 16 ft/s. Determine the time (in s) at which the mass passes through the equilibrium position. (Useg = 32 ft/s? for the acceleration due to gravity.) Find the time (in s) after the mass passes through the equilibrium position at which the mass attains its extreme displacement from the equilibrium position. What is the position (in ft) of the mass at this instant? ft

Chapter 3 Solutions

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the time at which the mass attains its extreme displacement from the equilibrium position and the position of the mass at this instant.

Solution Summary: The author calculates the time at which a mass attains its extreme displacement from the equilibrium position and the position of the mass at this instant.

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To find: the time at which the mass attains its extreme displacement from the equilibrium position and the position of the mass at this instant.

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Chapter 3 Solutions