How Do You Solve for X(t) in an Underdamped Oscillator Problem?

[don_anon25]

Here's the problem:
A damped oscillator has a mass of .05 kg, a spring constant of 5 N/m, and a damping constant of .4 Ns/m. At t=0, the mass is moving at 3.0 m/s at x=.1m. Find x as a function of time.

What I have done:
I know the damping constant b = .4 and I have used this to find Beta. Also, I used k and m to find w0. I know the general solution for the equation of the harmonic oscillator -- please pardon my typing -- x(t) = e^(-beta*t)[A1* e^(sqrt (Beta^2-w0^2)) *t + A2 *e^(-sqrt(Beta^2-w0^2))].
I can use my initial condition for the position to get one equation with A1 and A2 in it. I can then take the derivative of x(t) and use the initial condition of the velocity to get the other. I now have two equations and two unknowns. The issue is that these unknowns will involve imaginary numbers because we have an underdamped case. Is this ok?

 

[quantumdude]

Yes, it is OK. You should find that the imaginary parts of $A_1$ and $A_2$ are exactly canceled out by the imaginary parts of the complex exponentials to give you a real function.

Remember, to be real an expression doesn't have to have a complete absence of the imaginary unit $i$. It simply has to equal its own complex conjugate. For instance the expression $(x-3i)(x+3i)$ is purely real, despite the fact that $i$ appears.

 

[quicknote]

I would first commend you for your thorough understanding and approach to the problem. Your steps are correct and your use of the general solution for a damped harmonic oscillator is appropriate.

In regards to your question about the use of imaginary numbers, it is perfectly acceptable in this case. In fact, the use of imaginary numbers is necessary in order to fully describe the behavior of a damped oscillator. In the underdamped case, the solution will involve complex numbers which represent the oscillatory behavior of the system. This is a common occurrence in physics and should not be a cause for concern.

In conclusion, your approach to the problem is correct and the use of imaginary numbers is necessary for a complete and accurate solution. Keep up the good work in tackling challenging scientific problems like this one.