# Setting up equations for a vertical spring-mass system [closed]

I am preparing for an exam in classical mechanics and today I have encountered a slight problem when solving a more complex spring-mass system. Consider the following setup:

Then the system of equations should be

\begin{eqnarray} m_1g & = & m_1g-k(z_1-L)-k(z_1-z_2+L) \\ m_2g & = & m_2g-k(z_2-z_1-L) \end{eqnarray}

where $L$ is the natural length of the two springs. My solution was identical with the solution given, however, there was one small difference - I wrote the gravitational force in the first equation as $(m_1+m_2)g$ rather than simply $m_1g$.

Is there an error in my reasoning? The gravitational force on the first mass should involve the sum of the two masses...

## closed as unclear what you're asking by sammy gerbil, Gert, user36790, knzhou, garypJul 21 '16 at 3:08

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• Do you mean $m_1a = \ldots$ ?? – garyp Jul 20 '16 at 19:11
• Check the sign/direction of the force from the lower spring on $m_1$ in your 1st eqn. – sammy gerbil Jul 20 '16 at 19:15

Yes, there is an error in your reasoning. The gravitational force on an object is always proportional to the mass of that object. So if you want the gravitational force on $m_1$, you use $m_1 g$. If you want the gravitational force on $m_2$, you use $m_2 g$. If you want the gravitational force on the combined system of $m_1$ and $m_2$, you use $(m_1 + m_2)g$, but that wouldn't make much sense unless they're stuck together.