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1. The height of an object of mass 1 kg suspended on a spring whose spring constant is 5 N/m (when its support is in its rest position) is r(t). The resisting force on the object is directly proportional to the velocity of the mass, i.e. cx'(t), where c = 4 Ns/m. If y(t) is the vertical displacement of the support above its rest position, the equation for the motion of the mass is: x" +4x' +5x = 10 y(t) (a) Show that the Laplace transform X(s) = L{x(t)} of the vertical displacement of the mass for general initial conditions x(0) and x'(0) is given by, X (s) = x'(0) + (s+4)x(0) + 10Y(s) s² + 4s + 5 where Y(s) = L{y(t)} is the Laplace transform of the vertical displacement of the support. (b) Assume the mass is initially at rest, x(0) = x'(0) = 0, and the support is moved vertically upwards as y(t) = et. Find Y(s) and X(s). (c) Express X(s) in the following form: X (s): = A B(s+ 2) + C + s 1 (s + 2)² + 1 where A, B, and C are to be determined. (d) Find the displacement of the mass x(t).

1. The height of an object of mass 1 kg suspended on a spring whose spring constant is 5 N/m (when its support is in its rest position) is r(t). The resisting force on the object is directly proportional to the velocity of the mass, i.e. cx'(t), where c = 4 Ns/m. If y(t) is the vertical displacement of the support above its rest position, the equation for the motion of the mass is: x" +4x' +5x = 10 y(t) (a) Show that the Laplace transform X(s) = L{x(t)} of the vertical displacement of the mass for general initial conditions x(0) and x'(0) is given by, X (s) = x'(0) + (s+4)x(0) + 10Y(s) s² + 4s + 5 where Y(s) = L{y(t)} is the Laplace transform of the vertical displacement of the support. (b) Assume the mass is initially at rest, x(0) = x'(0) = 0, and the support is moved vertically upwards as y(t) = et. Find Y(s) and X(s). (c) Express X(s) in the following form: X (s): = A B(s+ 2) + C + s 1 (s + 2)² + 1 where A, B, and C are to be determined. (d) Find the displacement of the mass x(t).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. The height of an object of mass 1 kg suspended on a spring whose spring constant is 5 N/m (when its support is in its rest position) is x(t). The resisting force on the object is directly proportional to the velocity of the mass, i.e. cx'(t), where c = 4 Ns/m. If y(t) is the vertical displacement of the support above its rest position, the equation for the motion of the mass is: x" +4x' +5 x = 10 y(t) (a) Show that the Laplace transform X(s) = L{x(t)} of the vertical displacement of the mass for general initial conditions x(0) and x'(0) is given by, X(s) = x'(0) + (s+4)x(0) + 10Y (s) s² + 4s + 5 where Y(s) = L{y(t)} is the Laplace transform of the vertical displacement of the support. (b) Assume the mass is initially at rest, x(0) = x'(0) = 0, and the support is moved vertically upwards as y(t) = et. Find Y(s) and X(s). (c) Express X(s) in the following form: X (s) = A s-1 where A, B, and C are to be determined. (d) Find the displacement of the mass x(t). + B(s+ 2) + C (s + 2)² + 1
Transcribed Image Text:1. The height of an object of mass 1 kg suspended on a spring whose spring constant is 5 N/m (when its support is in its rest position) is x(t). The resisting force on the object is directly proportional to the velocity of the mass, i.e. cx'(t), where c = 4 Ns/m. If y(t) is the vertical displacement of the support above its rest position, the equation for the motion of the mass is: x" +4x' +5 x = 10 y(t) (a) Show that the Laplace transform X(s) = L{x(t)} of the vertical displacement of the mass for general initial conditions x(0) and x'(0) is given by, X(s) = x'(0) + (s+4)x(0) + 10Y (s) s² + 4s + 5 where Y(s) = L{y(t)} is the Laplace transform of the vertical displacement of the support. (b) Assume the mass is initially at rest, x(0) = x'(0) = 0, and the support is moved vertically upwards as y(t) = et. Find Y(s) and X(s). (c) Express X(s) in the following form: X (s) = A s-1 where A, B, and C are to be determined. (d) Find the displacement of the mass x(t). + B(s+ 2) + C (s + 2)² + 1
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