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In the third FBD, they made the whole system as one object. What happens if I don't want to do that? What if I jsut apply Newton's second Law to the mass M? I tried it out but it didn't work. Could someone point out my mistake? Note that my other force diagram are like the ones in the solutions. Note that $n_1'$ and $n_2'$ are the reaction forces for mass $m_1$ and $m_2$ respectively.


Writing out Newton's Second law for all the other massses

For M

$\begin{cases} \sum F_x = F - n'_2 = Ma \\ \sum F_y = n = n_1' + Mg + m_2g \end{cases}$

For $m_1$

$\begin{cases} \sum F_x = T = m_1 a \\ \sum F_y = n = m_1g \end{cases}$

For $m_2$

$\begin{cases} \sum F_x = n_2 = m_2 a \\ \sum F_y = T = m_2g \end{cases}$

If I eliminate the system of equations, I would get something absurd like $F = (M + m_2)a$

Could someoene please help me nail down this concept? I really appreciate it.

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You're not following the definition of Newton's law correctly... the FBD is of the single mass M, so there should be no gravitational force $m_2g$!! Anyway, you can only apply this law to a body, and there are are a finite amount of bodies here (including packaging multiple bodies into one)... you will see that you won't get very far in some of these cases. –  Chris Gerig Dec 27 '11 at 4:15
The problem is that your force is not just applied to M, it induces an acceleration which is distributed across the total mass... it's like you want to push a stack of books, but you instead want to ignore all books besides one of them; it does you no good. –  Chris Gerig Dec 27 '11 at 4:20
So it's impossible? I had $m_2g$ in there because doesn't M support both $m_1$ and $m_2$? –  sidht Dec 27 '11 at 4:27
Could I also ask again on the $m_2$'s motion? Remember when I asked about its vertical motion being 0 and you convinced me that it has horizontal motion with respect to the cart? Something is still off. When the mass on top accelerates forward, doesn't the string "shorten" on the top and therefore the string "below" attached to mass on pulley "extends"? And doesn't that mean there is in fact motion in the vertical direction? –  sidht Dec 27 '11 at 4:29
I want you to come to the answer yourself. When you apply Newton's Laws to a body you only consider the forces that are in immediate contact with it. –  Chris Gerig Dec 27 '11 at 4:30
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