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).
Chapter3: Polynomial Functions
Section3.5: Mathematical Modeling And Variation
Problem 7ECP: The kinetic energy E of an object varies jointly with the object’s mass m and the square of the...
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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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