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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) = cı 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). %3! • Identify the amplitude, frequency and phase for this simple harmonic motion and sketch

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) = cı 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). %3! • Identify the amplitude, frequency and phase for this simple harmonic motion and sketch

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Linear Methods

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 dy v(t) = y'(t) = dt dy a(t) = y"(t) = dt2 and 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. %3D dy 2 dt2 which has general solution y(t) = c1 cos(4t) + c2 sin(4t). + 32y(t) = 0 • 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). %3D • Identify the amplitude, frequency and phase for this simple harmonic motion and sketch a graph of y(t).
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 dy v(t) = y'(t) = dt dy a(t) = y"(t) = dt2 and 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. %3D dy 2 dt2 which has general solution y(t) = c1 cos(4t) + c2 sin(4t). + 32y(t) = 0 • 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). %3D • Identify the amplitude, frequency and phase for this simple harmonic motion and sketch a graph of y(t).
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