Explanation Formula used: The differential equation of motion calculated as: m d 2 y d t 2 + δ d y d t + k y = 0 . The Hooks Law is as: F = k x . Theorem used: “If r 1 = α + i β and r 2 = α − i β are the two complex roots of the auxiliary equation a r 2 + b r + c = 0 , then y = e α x ( c 1 cos β x + c 2 sin β x ) is the general solution of a y ″ + b y ′ + c y = 0 ”. Calculation: A weight( w 1 ) stretches first spring 3inch it means the displacement( x 1 ) of spring is 1 4 lb/ft and the instantaneous velocity δ 1 = 0 ft/sec . The weight is equal to the force of the object in this problem. It means F = m 1 g . From the Hooks Law F = k 1 x 1 Compare the value of F and substitute the value x 1 to calculate the value of spring constant. k 1 . 32 m 1 = k 1 × 1 4 k 1 = 128 m 1 So, the differential equation governing the problem as follows: m 1 d 2 y d t 2 + δ 1 d y d t + k 1 y = 0 . Substitute the value of m 1 , δ 1 and k 1 in the above equation. m 1 d 2 y d t 2 + 0 d y d t + 128 m 1 y = 0 m 1 d 2 y d t 2 + 128 m 1 y = 0 The boundary conditions of the problem is y ( 0 ) = 1 12 . So, the characteristics equation is as follows: m 1 r 2 y + 128 m 1 y = 0 r 2 + 128 = 0 On solving the above equation for calculating the value of r. r 2 + 128 = 0 r 2 = − 128 Squaring on both side in the above equation is as follows: r = 0 ± 8 2 i So, the general solution of differential equation is: y ( t ) = e 0 t ( c 1 cos 8 2 t + c 2 sin 8 2 t ) = c 1 cos 8 2 t + c 2 sin 8 2 t Substitute the boundary condition y ( 0 ) = 1 12 in y ( t ) = c 1 cos 8 2 t + c 2 sin 8 2 t . y ( 0 ) = c 1 cos 8 2 × 0 + c 2 sin 8 2 × 0 c 1 = 1 12 Differentiate the equation y ( t ) = c 1 cos 8 2 t + c 2 sin 8 2 t with respect to t . d d t y ( t ) = d d t ( c 1 cos 8 2 t + c 2 sin 8 2 t ) y ′ ( t ) = − 8 2 c 1 sin 8 2 t + 8 2 c 2 cos 8 2 t Substitute the boundary condition y ′ ( 0 ) = 0 f t / sec and value of c 1 in y ′ ( t ) = − 8 2 c 1 sin 8 2 t + 8 2 c 2 cos 8 2 t y ′ ( 0 ) = − 8 2 c 1 sin 8 2 t + 8 2 c 2 cos 8 2 t 0 = − 8 2 1 12 sin 8 2 × 0 + 8 2 c 2 cos 8 2 × 0 8 2 c 2 = 0 c 2 = 0 Substitute the value of c 1 and c 2 in. y ( t ) = c 1 cos 8 2 t + c 2 sin 8 2 t y ( t ) = 1 12 cos 8 2 t + 0 × sin 8 2 t = 1 12 cos 8 2 t For second spring: A weight( w 2 ) stretches first spring 9inch it means the displacement ( x 2 ) of spring is 3 4 lb/ft and the instantaneous velocity δ 2 = 0 ft/sec

Pearson eText for Thomas' Calculus: Early Transcendentals -- Instant Access (Pearson+)

14th Edition

ISBN: 9780137399185

Author: Joel Hass, Christopher Heil

Publisher: PEARSON+

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Chapter 17.3, Problem 14E

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The first time after t=0 when velocity of both the springs are equal in the given problem.

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Chapter 17.3, Problem 13E

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Chapter 17 Solutions

Pearson eText for Thomas' Calculus: Early Transcendentals -- Instant Access (Pearson+)

Ch. 17.1 - In Exercises 1−30, find the general solution of...Ch. 17.1 - In Exercises 1 – 30, find the general solution of...Ch. 17.1 - Prob. 3E Ch. 17.1 - Prob. 4E Ch. 17.1 - Prob. 5E Ch. 17.1 - Prob. 6E Ch. 17.1 - Prob. 7E Ch. 17.1 - Prob. 8E Ch. 17.1 - Prob. 9E Ch. 17.1 - Prob. 10E

Ch. 17.1 - Prob. 11E Ch. 17.1 - Prob. 12E Ch. 17.1 - Prob. 13E Ch. 17.1 - Prob. 14E Ch. 17.1 - Prob. 15E Ch. 17.1 - Prob. 16E Ch. 17.1 - Prob. 17E Ch. 17.1 - Prob. 18E Ch. 17.1 - Prob. 19E Ch. 17.1 - Prob. 20E Ch. 17.1 - Prob. 21E Ch. 17.1 - Prob. 22E Ch. 17.1 - Prob. 23E Ch. 17.1 - Prob. 24E Ch. 17.1 - Prob. 25E Ch. 17.1 - Prob. 26E Ch. 17.1 - Prob. 27E Ch. 17.1 - Prob. 28E Ch. 17.1 - Prob. 29E Ch. 17.1 - Prob. 30E Ch. 17.1 - Prob. 31E Ch. 17.1 - Prob. 32E Ch. 17.1 - Prob. 33E Ch. 17.1 - Prob. 34E Ch. 17.1 - Prob. 35E Ch. 17.1 - Prob. 36E Ch. 17.1 - Prob. 37E Ch. 17.1 - Prob. 38E Ch. 17.1 - Prob. 39E Ch. 17.1 - Prob. 40E Ch. 17.1 - Prob. 41E Ch. 17.1 - Prob. 42E Ch. 17.1 - Prob. 43E Ch. 17.1 - Prob. 44E Ch. 17.1 - Prob. 45E Ch. 17.1 - Prob. 46E Ch. 17.1 - Prob. 47E Ch. 17.1 - Prob. 48E Ch. 17.1 - Prob. 49E Ch. 17.1 - Prob. 50E Ch. 17.1 - Prob. 51E Ch. 17.1 - Prob. 52E Ch. 17.1 - Prob. 53E Ch. 17.1 - Prob. 54E Ch. 17.1 - Prob. 55E Ch. 17.1 - In Exercises 56−60, solve the initial value...Ch. 17.1 - Prob. 57E Ch. 17.1 - Prob. 58E Ch. 17.1 - Prob. 59E Ch. 17.1 - Prob. 60E Ch. 17.1 - Prob. 61E Ch. 17.1 - Prob. 62E Ch. 17.1 - Prob. 63E Ch. 17.1 - Prob. 64E Ch. 17.1 - Prob. 65E Ch. 17.1 - Show that if a, b, and c are positive constants,...Ch. 17.2 - Prob. 1E Ch. 17.2 - Prob. 2E Ch. 17.2 - Prob. 3E Ch. 17.2 - Prob. 4E Ch. 17.2 - Prob. 5E Ch. 17.2 - Prob. 6E Ch. 17.2 - Prob. 7E Ch. 17.2 - Prob. 8E Ch. 17.2 - Prob. 9E Ch. 17.2 - Prob. 10E Ch. 17.2 - Prob. 11E Ch. 17.2 - Solve the equations in Exercises 1−16 by the...Ch. 17.2 - Prob. 13E Ch. 17.2 - Prob. 14E Ch. 17.2 - Prob. 15E Ch. 17.2 - Prob. 16E Ch. 17.2 - Prob. 17E Ch. 17.2 - Prob. 18E Ch. 17.2 - Prob. 19E Ch. 17.2 - Prob. 20E Ch. 17.2 - Prob. 21E Ch. 17.2 - Prob. 22E Ch. 17.2 - Prob. 23E Ch. 17.2 - Prob. 24E Ch. 17.2 - Prob. 25E Ch. 17.2 - Prob. 26E Ch. 17.2 - Prob. 27E Ch. 17.2 - Prob. 28E Ch. 17.2 - Prob. 29E Ch. 17.2 - Prob. 30E Ch. 17.2 - Prob. 31E Ch. 17.2 - Prob. 32E Ch. 17.2 - Prob. 33E Ch. 17.2 - Prob. 34E Ch. 17.2 - Prob. 35E Ch. 17.2 - Prob. 36E Ch. 17.2 - Prob. 37E Ch. 17.2 - Prob. 38E Ch. 17.2 - Prob. 39E Ch. 17.2 - Prob. 40E Ch. 17.2 - Prob. 41E Ch. 17.2 - Prob. 42E Ch. 17.2 - Prob. 43E Ch. 17.2 - Prob. 44E Ch. 17.2 - Prob. 45E Ch. 17.2 - Prob. 46E Ch. 17.2 - Prob. 47E Ch. 17.2 - Prob. 48E Ch. 17.2 - Prob. 49E Ch. 17.2 - Prob. 50E Ch. 17.2 - Prob. 51E Ch. 17.2 - Prob. 52E Ch. 17.2 - In Exercises 53–58, verify that the given function...Ch. 17.2 - In Exercises 53–58, verify that the given function...Ch. 17.2 - Prob. 55E Ch. 17.2 - Prob. 56E Ch. 17.2 - Prob. 57E Ch. 17.2 - Prob. 58E Ch. 17.2 - Prob. 59E Ch. 17.2 - Prob. 60E Ch. 17.3 - Prob. 1E Ch. 17.3 - Prob. 2E Ch. 17.3 - A 20-lb weight is hung on an 18-in. spring and...Ch. 17.3 - Prob. 4E Ch. 17.3 - Prob. 5E Ch. 17.3 - Prob. 6E Ch. 17.3 - Prob. 7E Ch. 17.3 - Prob. 8E Ch. 17.3 - Prob. 9E Ch. 17.3 - Prob. 10E Ch. 17.3 - Prob. 11E Ch. 17.3 - Prob. 12E Ch. 17.3 - Prob. 13E Ch. 17.3 - Prob. 14E Ch. 17.3 - Prob. 15E Ch. 17.3 - Prob. 16E Ch. 17.3 - Prob. 17E Ch. 17.3 - Prob. 18E Ch. 17.3 - Prob. 19E Ch. 17.3 - Prob. 20E Ch. 17.3 - Prob. 21E Ch. 17.3 - Prob. 22E Ch. 17.3 - Prob. 23E Ch. 17.3 - Prob. 24E Ch. 17.3 - Suppose L = 10 henrys, R = 10 ohms, C = 1/500...Ch. 17.3 - Prob. 26E Ch. 17.4 - Prob. 1E Ch. 17.4 - Prob. 2E Ch. 17.4 - Prob. 3E Ch. 17.4 - Prob. 4E Ch. 17.4 - Prob. 5E Ch. 17.4 - Prob. 6E Ch. 17.4 - Prob. 7E Ch. 17.4 - Prob. 8E Ch. 17.4 - Prob. 9E Ch. 17.4 - Prob. 10E Ch. 17.4 - Prob. 11E Ch. 17.4 - Prob. 12E Ch. 17.4 - Prob. 13E Ch. 17.4 - Prob. 14E Ch. 17.4 - Prob. 15E Ch. 17.4 - Prob. 16E Ch. 17.4 - Prob. 17E Ch. 17.4 - Prob. 18E Ch. 17.4 - Prob. 19E Ch. 17.4 - Prob. 20E Ch. 17.4 - Prob. 21E Ch. 17.4 - Prob. 22E Ch. 17.4 - Prob. 23E Ch. 17.4 - Prob. 24E Ch. 17.4 - Prob. 25E Ch. 17.4 - Prob. 26E Ch. 17.4 - Prob. 27E Ch. 17.4 - Prob. 28E Ch. 17.4 - Prob. 29E Ch. 17.4 - Prob. 30E Ch. 17.5 - Prob. 1E Ch. 17.5 - Prob. 2E Ch. 17.5 - Prob. 3E Ch. 17.5 - Prob. 4E Ch. 17.5 - Prob. 5E Ch. 17.5 - Prob. 6E Ch. 17.5 - Prob. 7E Ch. 17.5 - Prob. 8E Ch. 17.5 - Prob. 9E Ch. 17.5 - Prob. 10E Ch. 17.5 - Prob. 11E Ch. 17.5 - Prob. 12E Ch. 17.5 - Prob. 13E Ch. 17.5 - Prob. 14E Ch. 17.5 - Prob. 15E Ch. 17.5 - Prob. 16E Ch. 17.5 - Prob. 17E Ch. 17.5 - Prob. 18E

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Chapter 17 Solutions