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Introductory Physics Homework Help
How do you solve for A in a critically damped oscillator problem?
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[QUOTE="derravaragh, post: 4321691, member: 458047"] [h2]Homework Statement [/h2] (A) A damped oscillator is described by the equation m x′′ = −b x′− kx . What is the condition for critical damping? Assume this condition is satisfied. (B) For t < 0 the mass is at rest at x = 0. The mass is set in motion by a sharp impulsive force at t = 0, so that the velocity is v0 at time t = 0. Determine the position x(t) for t > 0. (C) Suppose k/m = (2π rad/s)2 and v0=10 m/s. Plot, by hand, an accurate graph of x(t). Use graph paper. Use an appropriate range of t. [h2]Homework Equations[/h2] For critically damped, β[SUP]2[/SUP] = w[SUB]0[/SUB][SUP]2[/SUP] where β = b/(2m) and w[SUB]0[/SUB] = √(k/m) [h2]The Attempt at a Solution[/h2] Ok, for this problem, what I did initially was find the general form of position for a critically damped oscillator, which is: x(t) = (A + B*t)*e[SUP]-β*t[/SUP] and the velocity function is: v(t) = -Aβe[SUP]-βt[/SUP] + (Be[SUP]-βt[/SUP] - Bβte[SUP]-βt[/SUP]) Using the conditions given, I found: x(0) = A (obviously) which we don't know x(0) B = v[SUB]0[/SUB] + Aβ and x(t) can be rewritten as: x(t) = A(e[SUP]-βt[/SUP] + βte[SUP]-βt[/SUP]) + v[SUB]0[/SUB]te[SUP]-βt[/SUP] This is where I run into a wall. I can't seem to solve for A. I believe that x(0) should also be the max displacement since there is no driver for the impulse force, so A should be the max displacement, but this doesn't seem to get me anywhere. Any help on solving for A? I know how to do the rest other than that. [/QUOTE]
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How do you solve for A in a critically damped oscillator problem?
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