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I do not understand completely how do rotating pulleys with ropes (that do not slip) work. Besides the equations of motions, which are clear to me, some conditions on the accelerations are needed to solve the system with the equations. I'm having troubles in finding these conditions. Consider the following situations.

enter image description here

Here, since the rope doesn't slip, I wrote that:

$\begin{cases} \ddot{y_3}=R_2 \ddot{\theta}_2 \\ R_2 \ddot{\theta}_2=R_1 \ddot{\theta}_1 \\ \ddot{y_1}=R_1 \ddot{\theta}_1 \end{cases}$

But I don't think that this is right. In particular pulley 1 is accelerating downward, so maybe in the second equation I should add $\ddot{y_1}$ too. Furthermore I'm not sure at all about all the plus or minus that must be considered in the equations that I wrote. Is this the right way to solve the problem?

Then there are other cases with things like "yo-yo's". enter image description here

Here since the rope is getting longer but it does not slip the following equation must be always satisfied .

$x+y= l_0 +R_1 \theta_1+R_2 \theta_2\implies \ddot{x}+\ddot{y}=R_1 \ddot{\theta}_1+R_2 \ddot{\theta}_2$

But this doesn't help a lot and I tried to find other equations.

$\begin{cases} a_1= \ddot{y}= R \ddot{\theta}+R_1 \ddot{\theta}_1\\ a_2=\ddot{x}= - R \ddot{\theta}+R_2 \ddot{\theta}_2\end{cases}$

But again I'm not sure about signs, since in this way I'm assuming that the central pulley is rotating clockwise or counterclockwise, while that is unknown.

In general is there a way or a rule to follow to write such equations? Moreover is this the right way to think about the problem? i.e. that the objects and the points of the pulley in contact with ropes must have the same acceleration of ropes?

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closed as off-topic by Bill N, ACuriousMind, Gert, Kyle Kanos, CuriousOne Mar 19 '16 at 9:36

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In a problem like this, the starting point is always the analysis of the kinematics of the motion. The key to doing this is recognizing that the ropes are inextensible, and do not change in length. In your figure, just imagine that M3 moves down a distance x (so that the distance between the upper pulley and M3 increases by x). So the part of the rope on the right increases in length by x. This means that the two segments of rope to the left must decrease in total length by x. But the segments to the left are folded around the lower pulley, so the length of each of these decreases by x/2. So the upward acceleration of the left pulley is half the downward acceleration of M3.

The angular accleration of the right pulley is the downward acceleration of M3 divided by R2. The angular acceleration of the left pulley is the upward acceleration of this pulley divided by R1.

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But I don't think that this is right. In particular pulley 1 is accelerating downward, so maybe in the second equation I should add y1¨y1¨ too. Furthermore I'm not sure at all about all the plus or minus that must be considered in the equations that I wrote. Is this the right way to solve the problem?

If Pulley 1 is accelerating downward-

  1. one can start with writing length of the rope equals to the segments added together along with the part of the rope wound around the two pulleys.

If one differentiates the above relation twice with respect to time ,one can easily get a relation between accelerations of the pulley 1 and the mass M3.

  1. draw a free body diagram each for the pulley M1 (moving downward) and the mass M3. So, one gets the correct equations of motion.

  2. i think then the motion can be analyzed easily in detail using the interrelations of the translation - Regarding the rotation of the pulleys and the rotational energy -that can contribute to work done and an energy transfer picture and that can be done separately ,after you find out the individual accelerations.

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