Damped oscillator given odd initial conditions

[oddjobmj]

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 v₀ at time t=0. Determin the position x(t) for t>0.

(C) Suppose k/m=(2*pi rad/s)² and v₀=10 m/s. Plot an accurate graph of x(t) using an appropriate range for t.

Homework Equations

For critical damping; B=W₀

General solution for critically damped oscillator:

x(t)=(C₁+C₂t)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 b²=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(C₂-C₂βt-C₁β)

I know that at t=0 v(0)=v₀ so I can solve the solution for v(t) for a constant:

C₂=v₀+C₁β

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!

 

[ehild]

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!

The initial conditions are x(0)=0 and v(0)=vo. Impulsive force means that the mass is set into motion in an infinitesimally short time, so the displacement during that time is negligible.

ehild

 

[oddjobmj]

The initial conditions are x(0)=0 and v(0)=vo. Impulsive force means that the mass is set into motion in an infinitesimally short time, so the displacement during that time is negligible.

ehild

Ahh, fantastic. That definitely makes sense, thank you.

So, C₁=0 and C₂=v₀ making the general solution:

x(t)=v₀te^-βt

(C) Since k/m=(2*π rad/s)² and w₀²=k/m and w₀=β

β=2π rad/s

x(t)=10te^-2πt

The plot:
http://www.wolframalpha.com/input/?i=Plot(x(t)=10*t*e^(-2*pi*t)),+(t,+0,+1)

Thanks again!