- #1
Theorem.
- 237
- 5
Homework Statement
A critically damped oscillator with natural frequency [tex]\omega[/tex] starts out at position [tex]x_0>0[/tex]. What is the maximum initial speed (directed towards the origin) it can have and not cross the origin?
Homework Equations
For the case of critical damping,
[tex]x(t)=e^{(-\gammat)}(A+Bt) where \gamma=\omega=\sqrt{k/m}[/tex]
The Attempt at a Solution
Well first I derived the above equation (and verified it with my textbook). I then evaluated the initial position:
[tex]x(0)=A[/tex].
I then took the derivative of the position function to get velocity:
[tex]v(t)=(e^{-\gamma t}) ( B-Bt\gamma -A\gamma)[/tex]
Setting t=0 I obtained
[tex]v(0)=B-A\gamma[/tex] recalling that x0=A, [tex]v(0)= B-x_0\gamma[/tex]. I then solved for Beta: [tex] B=v_0 +\gamma x_0. [/tex]
I then tried substituting this back into the position equation, and solving for the initial velocity, the program is there is always time dependency that i can't get rid of... what am i doing wrong? any advice would be much appreciated!
Thanks
Last edited: