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$ma = mg - cx$, where $x(0) = x_0 = 0$ is the position in which there is no tension in the rope. $dx/dt = v_0$ for $t = 0$; $v_0$ is a known constant.

The discriminant of the characteristic equation is negative, we have complex roots, and the solution is a linear combination of a $\sin()$, $\cos()$ and a constant.

But, intuitively, everyone knows that $m$ would "swing up and down", which, I think, is a sinusoidal movement (intuitively!).

What is the physical significance of a second sinusoidal function in the solution?

Motivation of the question: I am trying to calculate the maximum tension in the rope, through the maximum acceleration. However, I find it unusually difficult to compute $\mathrm{max}(c_1\sin(\omega t) + c_2\cos(\omega t))$.

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I LaTeX-ified your equations. If you're interested, you can click "edit" to see the source for what I did. – Colin McFaul May 20 '13 at 13:18

The second solution is there to allow for arbitrary start and stop times. Using standard trig identities you can convert an arbitrary linear combination of $\sin$ and $\cos$ into a time-displaced sinusoidal function: $$A\sin(\omega t)+B\cos(\omega t)=R\cos(\omega(t-t_0)),$$ where $R=\sqrt{A^2+B^2}$ and $\tan(\omega t_0)=A/B$.

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What is the physical significance of a second sinusoidal function in the solution?

The physical significance is that two initial conditions are required for a unique solution.

Ignoring the constant, the general solution for this system is:

$x(t) = x(0) \cos \omega t + \dfrac{v(0)}{\omega} \sin \omega t$

Where $x(0), v(0)$ are the initial position and velocity respectively.

For example, if the initial velocity is 0, then:

$x(t) = x(0) \cos \omega t$

On the other hand, if the initial position is zero,

$x(t) = \dfrac{v(0)}{\omega} \sin \omega t$

The frequency in the denominator may look odd but see that it is correct by evaluating the time derivative at $t = 0$

$\dot x(t) = \dfrac{v(0)}{\omega} \omega \cos \omega t \rightarrow \dot x(0) = v(0)$

Another way to look at this is, like Emilio Pisanty points out, we need both a magnitude and a phase of the sinusoid to completely describe the motion.

However, I find it unusually difficult to compute max(c1sin(ωt)+c2cos(ωt)).

You've taken the time derivative and set it to zero, correct?

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