Problem 6. The simple harmonic motion of a 2 kg mass attached to a spring with spring constant, k = 32, is governed by the differential equation. dy 2- + 32y(t) = 0 which has general solution dt2 y(t) = c1 cos(4t) +c2 sin(4t). • Show that y(t) = 5 cos(4t +) is a solution to this differential equation by expressing this function as a linear combination of y1(t) = cos(4t) and y2(t) = sin(4t). • Identify the amplitude, frequency and phase for this simple harmonic motion and sketch a graph of y(t).
Problem 6. The simple harmonic motion of a 2 kg mass attached to a spring with spring constant, k = 32, is governed by the differential equation. dy 2- + 32y(t) = 0 which has general solution dt2 y(t) = c1 cos(4t) +c2 sin(4t). • Show that y(t) = 5 cos(4t +) is a solution to this differential equation by expressing this function as a linear combination of y1(t) = cos(4t) and y2(t) = sin(4t). • Identify the amplitude, frequency and phase for this simple harmonic motion and sketch a graph of y(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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Question
Transcribed Image Text:Problems 6 though 10 involve modelling the motion of an object attached to
a spring. The function, y(t), denotes the displacement of the mass from the
equilibrium position, and is a function of time, t. The velocity and acceleration
are
v(t) = y'(t) =
dt
dy
and
ddy
a(t) = y"(t) =
dt2
respectively.
All work, solutions and answers should be written in terms of t.
Problem 6. The simple harmonic motion of a 2 kg mass attached to a spring with spring
constant, k = 32, is governed by the differential equation.
dy
2-
+ 32y(t) = 0
which has general solution
y(t) = c1 cos(4t) + c2 sin(4t).
dt2
• Show that y(t) = 5 cos(4t +) is a solution to this differential equation by expressing
this function as a linear combination of y1(t) = cos(4t) and y2(t) = sin(4t).
• Identify the amplitude, frequency and phase for this simple harmonic motion and sketch
a graph of y(t).
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