Superposition in classical Mechanics I watched a video in YouTube solved these problem using superposition 
First he fixed the bottom rope and found the accelearations 
Second he fixed the upper pulley and found the accelerations 
Third he added the results - as we do in solving AC steady state circuits - 
I wondered does it work because Newton's Law is linear? 
can I solve any classical mechanics problems involving many systems by that method or does it have limitations? 
The video: http://www.youtube.com/watch?v=LfNsGuR6rVA

 A: Consider the well-known mechanical harmonic oscillator, where you combine Newtons 2nd law $F = m\ddot{x}$ for the motion of an object with mass $m$, with Hooke's law $F = -kx$ for the force from a spring with stiffness $k$. The resulting differential equation is:
$$ m\ddot{x} + kx = 0$$
This is a linear differential equation, because if we add the equation for a position $x_1$ and a position $x_2$, then we obtain an equation of exactly the same form for $x_1+x_2$:
$$ (m\ddot{x}_1 + kx_1) =0\quad\text{and}\quad (m\ddot{x}_2 + kx_2) =0 \\ \Downarrow \\ a(m\ddot{x}_1 + kx_1) + b(m\ddot{x}_x + kx_x) = 0 \\ \Downarrow \\ m(a\ddot{x}_1+b\ddot{x}_2) + k(ax_1+bx_2) = 0$$
This is the source of the superposition principle: any linear combination $ax_1+bx_2$ of two solutions $x_1$ and $x_2$ to the differential equation, also has to obey the same differential equation. Thus, any superposition of solutions are also solutions.
However, Hooke's law is only valid for relatively small displacements $x$; in general, the force from a deformed material will be a nonlinear function, which varies between different kinds of materials. According to Wikipedia:

Hooke's law is only a first order linear approximation to the real response of springs and other elastic bodies to applied forces. It must eventually fail once the forces exceed some limit, since no material can be compressed beyond a certain minimum size, or stretched beyond a maximum size, without some permanent deformation or change of state. In fact, many materials will noticeably deviate from Hooke's law well before those elastic limits are reached.

Let us consider such an example. Let's say that we're studying a material where the spring force is $F=-\tanh(kx)$. As you can see from plotting the function, the hyperbolic tangent behaves nearly linearly for small $x$ and therefore obeys Hooke's law in this region. However, for large positive values of $x$, this force will converge towards $-1$, while for large negative values of $x$ it converges to $+1$. A physical motivation for considering such a spring force, would be that once e.g. a metallic spring is stretched or compressed to a certain threshold, any additional force you apply will simply deform the material, which reduces the spring restoration force. Thus, after a certain threshold, we expect a nearly constant spring force due to the deformation of the material.
Combining $F=m\ddot{x}$ with $F=-C\tanh(kx)$, we obtain the following differential equation:
$$ m\ddot{x} + C\tanh(kx) = 0 $$
This is an example of a nonlinear differential equation. If we try to add the differential equation for two solutions $x_1$ and $x_2$ like before, then we would obtain:
$$ m(\ddot{x}_1 + \ddot{x}_2) + C(a+b)\frac{e^{2k(x_1+x_2)} - 1}{(e^{2kx}+1)(e^{2ky}+1)} = 0 $$
So we see that in this case, $ax_1+ax_2$ follows an entirely different differential equation from $x_1$ and $x_2$, since $a\tanh(kx_1)+b\tanh(kx_2)\neq \tanh(akx_1+bkx_2)$. Therefore the superposition principle does not hold for this kind of system. 
