A mass of 24 pounds is attached to a spring and stretches it 4 inches. If the mass is released from rest (i.e. no initial velocity) at a position 3 inches above its equilibrium position, and if there is no damping, (a) Model this situation as a second order initial value problem. Explicitly determine the value of the spring constant. (b) Solve it to determine the position u(t) of the mass at any time t. (c) Determine the frequency, period, amplitude, and phase of motion. Formula for Real Valued Solutions of 2 × 2 systems of linear ODES with complex eigenvalues: [cos(ẞt)u— sin(ßt)w] x(t)=C₁eat cos(ẞt) u - sin(ẞt) w +C₂eat sin(ẞt)u+cos(ẞt) w w] where the system has eigenvalues λ = α ± ẞi and eigenvectors v = u+iw, where α, ẞ are real numbers, and u and w are real vectors.
A mass of 24 pounds is attached to a spring and stretches it 4 inches. If the mass is released from rest (i.e. no initial velocity) at a position 3 inches above its equilibrium position, and if there is no damping, (a) Model this situation as a second order initial value problem. Explicitly determine the value of the spring constant. (b) Solve it to determine the position u(t) of the mass at any time t. (c) Determine the frequency, period, amplitude, and phase of motion. Formula for Real Valued Solutions of 2 × 2 systems of linear ODES with complex eigenvalues: [cos(ẞt)u— sin(ßt)w] x(t)=C₁eat cos(ẞt) u - sin(ẞt) w +C₂eat sin(ẞt)u+cos(ẞt) w w] where the system has eigenvalues λ = α ± ẞi and eigenvectors v = u+iw, where α, ẞ are real numbers, and u and w are real vectors.
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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![A mass of 24 pounds is attached to a spring and stretches it 4 inches. If the mass is released
from rest (i.e. no initial velocity) at a position 3 inches above its equilibrium position, and if there
is no damping,
(a) Model this situation as a second order initial value problem. Explicitly determine the value of
the spring constant.
(b) Solve it to determine the position u(t) of the mass at any time t.
(c) Determine the frequency, period, amplitude, and phase of motion.
Formula for Real Valued Solutions of 2 × 2 systems of linear ODES
with complex eigenvalues:
[cos(ẞt)u— sin(ßt)w]
x(t)=C₁eat cos(ẞt) u - sin(ẞt) w +C₂eat
sin(ẞt)u+cos(ẞt) w
w]
where the system has eigenvalues λ = α ± ẞi and eigenvectors v = u+iw, where α, ẞ are real
numbers, and u and w are real vectors.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F183fa540-aa6e-4b2e-819b-0b2a4459f410%2F7dc0c87b-1a82-46b2-a25e-c7185b4d2d60%2Flslaps_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A mass of 24 pounds is attached to a spring and stretches it 4 inches. If the mass is released
from rest (i.e. no initial velocity) at a position 3 inches above its equilibrium position, and if there
is no damping,
(a) Model this situation as a second order initial value problem. Explicitly determine the value of
the spring constant.
(b) Solve it to determine the position u(t) of the mass at any time t.
(c) Determine the frequency, period, amplitude, and phase of motion.
Formula for Real Valued Solutions of 2 × 2 systems of linear ODES
with complex eigenvalues:
[cos(ẞt)u— sin(ßt)w]
x(t)=C₁eat cos(ẞt) u - sin(ẞt) w +C₂eat
sin(ẞt)u+cos(ẞt) w
w]
where the system has eigenvalues λ = α ± ẞi and eigenvectors v = u+iw, where α, ẞ are real
numbers, and u and w are real vectors.
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