# Pushing, inertia of a cart system

I have a cart with another cart on top which gets pulled down by another cart on the side. There is no friction.

The question is:

How strongly do I have to push with $F$ to keep the cart $m_1$ stable?

I work in the system of the cart $m_3$. This system accelerates with some $a$. The force of inertia pulls back on $m_1$ with $m_1 a$. That is opposed by the gravitational force of $m_2$, which is simply $m_2 g$.

The acceleration that I need for this to be stable is:

$$a = \frac{m_2}{m_1} g$$

Here comes the point I am not too sure about:

The force is on $m_3$ and on $m_2$, so the driving force would be this:

$$F = (m_2 + m_3) \frac{m_2}{m_1} g$$

On second thought, I think also need to accelerate $m_1$ some way or another, so that my total force would be a little higher:

$$F = (m_1 + m_2 + m_3) \frac{m_2}{m_1} g$$

I prefer the latter, but what is the right solution?

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By the way, inertia is not a force. You could consider it a fictitious force that appears in an accelerating reference frame, which I think may be what you've done here, but using fictitious forces can get you really confused if you don't understand what you're doing very well, so I generally recommend against using fictitious forces if you don't have to. And this problem can definitely be done without them. –  David Z Feb 5 '12 at 0:59
True. I think I translated all the coordinates, velocities and accelerations correctly, so this should work out. –  queueoverflow Feb 5 '12 at 16:21
The force acts not just on $m_3$ and $m_2$, it acts on $m_1$ as well (via the pulley), so I would prefer the second answer.