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## Homework Statement

Hi everybody! I'm doing a problem about oscillations, and I must admit that a few things are still unclear to me about that subject. Can someone maybe help me?

a) A onedimensional masspoint m is oscillating under the influence of the force [tex]F(x) = -c⋅x (c > 0).[/tex] What is the Newton's equation of motion for the system and integrate it with an exponential approach [tex]x(t) ∼ e^{λt}.[/tex] Find the integration constant for the initial conditions [tex]x(t = 0) = x_0[/tex] and [tex]\dot{x}(t = 0) = \dot{x_0}.[/tex]

b) A friction force [tex]-ϒ \dot{x} (ϒ > 0)[/tex] is added to the system of a), that means a total force [tex]F(x,\dot{x}) = -cx - ϒ\dot{x}[/tex] is acting upon the masspoint m. Similarly to question a), find the path [tex]x(t)[/tex] and distinguish it between the cases:

[tex] 0 < \frac{ϒ}{m} < 2ω_0 : \mbox{underdamping}[/tex]

[tex] \frac{ϒ}{m} = 2ω_0 : \mbox{critical damping}[/tex]

[tex] 0 < 2ω_0 < \frac{ϒ}{m} : \mbox{overdamping}[/tex]

## The Attempt at a Solution

a) So first I set up my equation of motion:

[tex] m \cdot \ddot{x} = -c \cdot x ⇔ m \cdot \ddot{x} + c \cdot x = 0 ⇔ \ddot{x} + \frac{c}{m} x = 0 \\

\mbox{Let } ω_0 = \sqrt{\frac{c}{m}} ⇒ \boldsymbol{\ddot{x} + ω_0^2 x = 0} [/tex]

Then I rewrite the equation in the exponential form and solve for λ:

[tex]x(t) = α \cdot e^{λt} ⇒ α \cdot λ^2 \cdot e^{λt} + α \cdot ω_0^2 \cdot e^{λt} = 0 ⇔ λ^2 + ω_0^2 = 0 \\

⇒ \boldsymbol{λ = ± i \cdot ω_0} ⇒ x(t) = α \cdot e^{i ω_0 t} + α^{*} \cdot e^{-i ω_0 t}

[/tex]

Then do I have to integrate the equation of motion by substituting x(t) by that expression? Why would we want to do that anyway? To get the velocity? It's the first time I'm ever asked to integrate an equation of motion, and there is no explanation whatsoever with the problem about why should one do that. Anyway I gave it a go:

[tex]\int -α \cdot ω_0^2 \cdot e^{i ω_0 t} + α \cdot ω_0^2 \cdot e^{-i ω_0 t} dt = -α \cdot ω_0^2 \int e^{i ω_0 t} dt+ α \cdot ω_0^2 \int e^{-i ω_0 t} dt \\

= i \cdot α \cdot ω_0 (e^{i ω_0 t} + \frac{1}{e^{-i ω_0 t}}) + C[/tex]

Ehem...Not sure what is that, if it's correct and what to do from there! I do not know what α is, and all those imaginary units kind of scare me! Some help/hint here would be very appreciated. :)

Thanks a lot in advance for your answers!

Julien.