# Differential Pulley - Calculation of Mechanical Advantage

In the above graphic, the four segments of the chain are labelled W, X, Y and Z. The magnitudes of their corresponding forces are FW, FX, FY and FZ, respectively.

Assuming that the chain is massless, FX = 0 because segment X is not supporting any weight.

Taking the system at equilibrium, FW and FY are equal — if they were not, the lower pulley would freely turn until they were.

Next, the downward force acting on the lower pulley equal the upward forces acting on it, so

FL = FW + FY, or 2 FW because FW = FY.

Additionally, there is no net torque or moment around the compound pulley, so the clockwise torque is equal to the anticlockwise torque:

FW R + FX r = FY r + FZ R .

Substituting FX and FY from the above equations,

FW R + 0 = FW r + FZ R .

Rearranging gives

FW = FZ · R/Rr .

As FW = F L/2,

F L/2 = FZ · R/Rr .

Finally, the mechanical advantage, F L/F Z = 2 R/Rr or 2/1 − r/R .

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