Query 21 Differential Equations

Question:  A 10 kg mass stretches a spring 9.8 cm beyond its original rest position. 

A driving force F(t) = 20 N * cos((8 s^-1) * t) begins at t = 0, where the downward direction is regarded as positive. 

Write down and solve the appropriate differential equation, obtaining the position function for the motion of the mass.

Plot your solution, and find the maximum distance of the mass from its equilibrium position.

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Question:  The motion of a mass is governed by the equation

m y '' + 2 gamma y ' + omega_0^2 y = F(t),

with m = 2 kg, gamma = 8 kg / s and k = 80 N / m and F(t) = 20 N * e^(- t s^-1).

Solve the equation for the function y(t).

What is the long-term behavior of this system?

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Question: 

Solve the equation

y '' + 2 delta y ' + omega_0^2 y = F cos( omega_1 * t), y(0) = 0, y ' (0) = 0.

Give an outline of your work.  A very similar problem was set up and partially solved in class on 110309, and your text gives the solution but not the steps.

Find the limiting function as omega_1 approaches omega_0, and discuss what this means in terms of a real system.

Find the limiting function as delta approaches 0, and discuss what this means in terms of a real system.

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Question:  An LC circuit with L = 1 Henry and C = 4 microFarads is driven by voltage V_S(t) = 10 t e^(-t).

Write and solve the differential equation for the system.

Interpret your result.

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