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So, let's consider a simple ideal pulley system where there's a weight to be pulled, and a counterweight on the other side

According to my understanding (feel free to nitpick), there's a restoring force opposing the weight of the box that keeps it balanced. When we apply a downward force to the rope, it increases the restoring force which is upwards, and as it goes along the rope and over the pulley, this force changes direction and becomes downwards, adding onto the weight of the counterweight, but then how does the box actually go upwards? The force should be pushing the counterweight downwards

I don't know to be frank, I'm missing something most probably

This is the best I got, and it's definitely incorrect:

Diagram

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There is no such thing as a "restoring force" keeping the two boxes balanced on the pulley; this is not a system like a simple pendulum where we have a change in potential energy that gives rise to the restoring force. The terminology is not appropriate here.

The boxes are balanced because their masses are equal, so the weight $W_2$ of the right box pulling on the left box is equal to the weight $W_1$ of the left box, and the weight $W_1$ of the left box pulling on the right box is equal to the weight $W_2$ of the right box. It is the right box that counteracts the weight of the left box, and vice versa, keeping the system in equilibrium (i.e. balanced). Mathematically, we can write the following. In the left box's POV $$ \sum F_1=W_2-W_1=m_1a=0 $$ while in the right box's POV: $$ \sum F_2=W_1-W_2=-m_2a = 0 $$

(The negative sign is there since if the left box moves down, the right box moves up.)

When you apply an external force by pulling the left box down, your force breaks the force equilibrium by directly moving the left box down. Your force also adds to the weight of the left box, producing a new total force directed upward in the right box's POV. This force is larger than the weight $W_2$ of the right box which is directed downward, thus causing the right box to move up.

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  • $\begingroup$ " increasing the tension T1 to become T1=W1+F which becomes larger than the weight W2 of the right box, thus causing it to move up" Could you expand on this part? How does T1 cause second object to move up? $\endgroup$
    – damacc
    Commented Jul 7 at 13:12
  • $\begingroup$ I found my previous explanation confusing due to the use of tension. In analyzing motion like this, there is no need to take tension into account, which may only complicate things (see this discussion). There should be no loss of information by doing this, unless you specifically care about the tension. $\endgroup$
    – hendlim
    Commented Jul 7 at 13:13
  • $\begingroup$ @damacc Before you apply the force, you have $W_1$ trying to pull the right box up, and $W_2$ trying to pull the right box down. But since they are equal in magnitude, they only cancel each other out, hence no motion. After you apply the force, you have $F$ which helps $W_1$ pull the right box up. Together, $F+W_1$ beats $W_2$. $F$ and $W_1$ pulls up stronger than $W_2$ pulls down, so the right box moves up. $\endgroup$
    – hendlim
    Commented Jul 7 at 13:17
  • $\begingroup$ Ah, so the upward tension T1 has a counteracting T1, and same for T2? $\endgroup$
    – damacc
    Commented Jul 7 at 13:25
  • $\begingroup$ "There should be no loss of information by doing this, unless you specifically care about the tension." Yeah I understand, but I do care to understand the whole picture (with tension) $\endgroup$
    – damacc
    Commented Jul 7 at 13:29

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