# How to Analyze Applied Forces/Torques of System With Multiple Massless/Frictionless Pulleys of Different Radii

I have been reviewing the basics of mechanics in preparation of studying Spivak's text book on mechanics (I am from a more mathematical background and I am taking an advanced analytical mechanics course next quarter which is using this text). I recently came across a problem in a text different from the one that I am using (University Physics), and I was having difficulty understanding how to approach it, and indeed even understanding the details of the situation (the problem is stated with little detail). Unfortunately, my text does not seem to cover anything like it, in the exercises or otherwise.

Imagine a flat surface inclined at 45 degrees. A 1500 kg block is placed on the incline, and it is (according to the figure) attached directly to a pulley of radius $r$. The pulley itself has a cable rolling over it which extends along the incline with one end connected to the surface and the other end connected to a larger pulley of radius $3r$, but at a point of contact which is only $r$ from the center. Another block of unknown mass $m$ is attached to the larger pulley at the edge ($3r$ from the center) and is allowed to hang. We are asked to compute the mass $m$ of the hanging block required to counteract the motion of the block of mass 1500 kg. (The physical situation [I will try to update this with a picture as soon as I can] is basically just like the traditional one pulley/two-block system on an incline first encountered with problems involving Newton's laws; except here there are two pulleys involved as described above).

To be perfectly honest, I am really not at all sure what is going on in this problem. If we take the axis of rotation to be the center of the larger pulley, then it is clear the second block imparts a torque of $3rmg\sin\left(\frac{\pi}{4}\right)$ and a tension on the cable is just $mg$. It seems to be the block on the incline imparts no torque since its action is directed parallel to the radial coordinate of the axis (if we are taking the natural coordinate system with axes perpendicular and parallel to the axis of rotation). What is the significance of the smaller pulley to which the mass is attached?

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