A pulley system may comprise of smaller component pulley systems. Take the case of a pulley system, which consists of two component pulley systems (“B” and “C”) as shown in the figure.

Mixed pulley system |
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The component pulley system, usually, is a simple arrangement of two blocks connected with a single string passing over the pulley. The arrangement is alike static pulley system except that the pulley (“B” or "C") is also moving along with other constituents of the component systems like string and blocks.

If we could treat the moving pulley system static, then the analysis for the motion of blocks would become very easy. In that case, recall that the accelerations of the blocks connected with single string has equal magnitudes of accelerations, which are oppositely directed. This sense of oppositely directed acceleration, however, is not valid with moving pulley. A block, which appears to have a downward acceleration with respect to static pulley, say 1

This simple framework of analysis involves two steps :

- Analyze for accelerations of blocks with respect to moving reference of pulley.
- Use concept of relative acceleration to determine accelerations of blocks with respect to ground.

For understanding the technique, we concentrate on one of the component systems that of pulley "B" as shown in the figure above. Let “

Applying concept of relative motion, we can expand relative accelerations as :

where accelerations on the right hand side of the equation are measured with reference to ground. Clearly, using these expansions, we can find accelerations of blocks with respect to ground as required.

** Example **

Problem :
In the arrangement shown the accelerations of blocks “1”, “2” and “3” are 2

Mixed pulley system |
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Solution : The accelerations of three blocks are given with respect to ground. For analysis in moving frames of pulleys, we assign directions of relative accelerations as shown in the figure below. The pulley “B” and “C” are hanging from fixed pulley, “A”. If pulley “B” moves up, then pulley “C” moves down. The blocks “1” and “2”, in turn, are hanging from moving pulley “B”. If relative acceleration of “1” is up, then relative acceleration of “2” is down as they are connected by one string. Similarly, the blocks “3” and “4” are hanging from moving pulley “C”. If relative acceleration of “3” is up, then relative acceleration of “4” is down as they are connected by one string.

Mixed pulley system |
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Let us now consider upward direction as positive. Since relative acceleration of block "1" is equal and opposite to relative acceleration of block "2", we have :

Expanding in terms of absolute accelerations, we have :

Putting given values,

Now, pulleys “B” and “C” are connected through a string passing over a static pulley “A”. The accelerations of the moving pulleys “B” and “C” are, therefore, equal in magnitude but opposite in direction. From the analysis above, the pulley “B” has upward acceleration of 5

Again following the earlier reasoning that relative accelerations are equal and opposite, we have :

Putting values,

The negative sign shows that the block “4” moves down i.e. opposite to the positive reference direction, which is upward.