- #1

oddjobmj

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## Homework Statement

(A) The damped oscillator is described by the equation mx''+bx'+kx=0. What is the condition for critical damping expressed in terms of m,b,k. Assume this is satisfied.

(B) For t<0 the mass is at rest (x=0). This mass is set in motion at t=0 by a sharp impulsive force so that the velocity is v

_{0}at time t=0. Determin the position x(t) for t>0.

(C) Suppose k/m=(2*pi rad/s)

^{2}and v

_{0}=10 m/s. Plot an accurate graph of x(t) using an appropriate range for t.

## Homework Equations

For critical damping; B=W

_{0}

General solution for critically damped oscillator:

x(t)=(C

_{1}+C

_{2}t)e

^{-βt}

## The Attempt at a Solution

I am running into issues at (B) where I'm not entirely sure how to find the values of the two constants in the general solution.

(A) I have shown through the given relations that b

^{2}=4km which I believe is the correct answer.

(B) Given the general solution for a critically damped oscillator I can take its derivative to find v(t):

v(t)=e

^{-βt}(C

_{2}-C

_{2}βt-C

_{1}β)

I know that at t=0 v(0)=v

_{0}so I can solve the solution for v(t) for a constant:

C

_{2}=v

_{0}+C

_{1}β

I'm just not sure about how to solve for x(t). Can I consider x(0)=0 since it is set in motion at t=0 but since no time has elapsed it has not moved?

If not, how do I proceed in solving for the constants? I believe the rest of the problem will fall into place without much issue after that point.

Thank you for your time and help!