Third System of Pulleys
It is quite identical to the first system; however by looking at the figure it becomes clear that the operations involved are just in the reverse process.
The velocity ratio of the system can be tracked by following a unit movement of the weight.
Suppose, the attached weight is moved by a distance of x meters by the effort applied at P, this will cause an instantaneous slackening of the strings involved.
For supporting the action, the strings will go through a sequential tightening movement through the pulley rotations.
Therefore the slackening of string 1 (assuming to be equal to x meters) is compensated by pulley number two, which comes down and covers a distance of 2x meters.
Also with slackening of string 2, x1 gets pulled across a distance of 2x –x = x meters.
Continuing further, with x1 being pulled through x meters, x2 gets pulled through a length of 2x +x = 3x = (22 – 1)x.
The procedure is followed on to keep the relative position of pulley 3 constant and string 3 is pulled across a distance of (2 × 3x × x) = 7x = (23 – 1)x, and finally string x4, which is actually the effort, crosses a distance of (2 × 7x + x) = 15x = (24 – 1)x meters.
Therefore the VR of the system can be equated as = Distance Covered by Effort/Distance Covered by Weight = (24 – 1)x/x = 24 – 1, for the present example which consists of 4 pulleys.
In general for a particular third system of pulley having n number of pulleys, VR = 2n – 1.
MA and ɳ may be taken as discussed for the previous systems.
Images Drawn by Swagatam, Courtesy - Applied Mechanics and Strength of Materials, By R.S. Khurmi