9.3 Working with moving pulleys

Analysis of motion with moving pulley relies on constraint relation and force analysis for moving entities like pulleys and blocks. This classic approach, however, has a serious problem. As there may be large numbers of blocks and other elements, the numbers of unknowns and corresponding numbers of available equations become very large and analysis of motion becomes very difficult.

Further, we consider acceleration of each movable part with reference to ground. Most of the time, it is difficult to guess the directional relationship among the accelerations of various entities of the system. In this module, we shall introduce few simplifying methods that allow us to extend the use of the simple framework of static pulley for analyzing motion of movable pulley.

Component pulley system

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.

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 $m/s^2$ , will have an upward acceleration of 2 $m/s^2$ with respect to ground as pulley itself may have an upward acceleration of 3 $m/s^2$ . This situation leads to a simplified framework for determining acceleration.

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 “ $a_1$ ” and “ $a_2$ ” denote accelerations of the two blocks with respect to ground reference. Let “ $a_B$ ” denotes the acceleration of pulley “B” with respect to ground. Also, let “ $a_{1B}$ ” and “ $a_{2B}$ ” denote relative accelerations of the two blocks with respect to moving pulley “B”. Analyzing motion of blocks with respect to moving reference of pulley, we have :

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.