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We see the mass-and-spring system above and we derive two systems of differential equations to describe its behavior.

For example, we have

$$m_2 y^{\prime \prime} = k_2 x - (k_2 + k_3)y$$

I have two questions:

1. Is there any formal method of deriving these differential equations? As I usually only rely on looking at the figure and deriving them myself.

2. For the equation aforementioned, why do we only include $k_2 x$? Wouldn't there be another force acting on the system caused by the stretching of the spring represented by $k_1$, which would add a $-k_1 x$ term to the equation? Or is that force somehow cancelled?

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    $\begingroup$ The equation for a block should only include forces from things touching that block. The first spring does not touch the second block, so the $k_1x$ term only shows up for the first block. But don't worry, as it moves the first block it pulls on the second spring, which will affect the second block. $\endgroup$ – btilly Apr 26 '16 at 1:23
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  1. There is a more formal way that takes a lot of guessing out: the Lagrange equations (formalism), and the Hamilton equations (the so-called analytical mechanics).

  2. When you write the motion for mass $2$, note that the only forces that act on it are through the two springs. All you need is to figure out the signs correctly. Namely: if the spring $k_3$ is elongated to the right of equilibrium position (compressed), it will act on mass $2$ towards the left of the axis $Oy$, hence $-k_3 y$. The spring $k_2$ is compressed by $y-x$ (contributing to the total force as $-k_2 (y-x)$ (imagine if you were replacing mass $2$ and $y>x$, which way you would feel the pull).

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