# Differential pulley

Example of a differential pulley

A differential pulley, also called "Weston differential pulley", sometimes "chain hoist" or colloquially "chain fall", is used to manually lift very heavy objects like car engines. It is operated by pulling upon the slack section of a continuous chain that wraps around pulleys. The relative size of two connected pulleys determines the maximum weight that can be lifted by hand.

## History

The differential pulley was invented in 1854 by Thomas Aldridge Weston from King's Norton, England.[1][2]

The pulleys were manufactured in collaboration with Richard and George Tangye. According to Richard Tangye's autobiography, the Weston differential pulley evolved from the Chinese windlass, with an endless chain replacing the finite length of rope. He claimed that many engineering firms conceded on the difficulty of efficiently disengaging the chain from the teeth as the pulleys turned, but his firm developed a "pitch" chain which solved the issue. Marketed as "Weston Differential Pulley Blocks with Patent Chain Guides", the pulley had good sales, namely, 3000 sets in 9 months. It was displayed in 5 sizes — from 10 long hundredweight (510 kg) to 3 long tons (3,000 kg) — at the 1862 International Exhibition in London and received a medal for "original application, practical utility and success".

An ironmonger challenged the Tangyes that the pulley had been in use for 30 years before Weston's patent but the judge, William Page Wood ruled in favour of the Tangyes because the engaging mechanism was substantially different from the one presented as evidence.[3]

The Yale Lock Company acquired the patent rights in 1876.[2]

## Mode of operation

A differential pulley can lift very large masses a short distance. It consists of two fixed pulleys of unequal radii that are attached to each other and rotate together, a single pulley bearing the load, and an endless rope looped around the pulleys. To avoid slippage, the rope is usually replaced by a chain, and the connected pulleys by sprockets.

The two sections of chain carrying the single pulley exert opposing and unequal torques on the connected pulleys, such that only the difference of these torques has to be compensated manually by pulling the loose part of the chain. This leads to a mechanical advantage: the force needed to lift a load is only a fraction of the load's weight. At the same time, the distance the load is lifted is smaller than the length of chain pulled by the same factor. This factor (the mechanical advantage MA) depends on the relative difference of the radii r and R of the connected pulleys:

$M\!A = \frac{2 R}{R-r} = \frac{2}{1 - \frac{r}{R}}$

The effect on the forces and distances (see figure) is quantitatively:

$F_\mathrm{Z}=\frac{F_\mathrm{L}}{M\!A}\quad,\quad h=\frac{s}{M\!A}\quad.$

The difference in radii can be made very small, making the mechanical advantage of this pulley system very large.[4][5] In the extreme case of zero difference in radii, MA becomes infinite, thus no force (besides friction) is needed to move the chain, but moving the chain will no longer lift the load.

At the other extreme, when r is zero, the system becomes a simple gun tackle with a mechanical advantage of 2.

Comparison of a differential pulley (left) and a differential windlass or Chinese windlass (right). The rope of the windlass is depicted as spirals for clarity, but is more likely helices with axes perpendicular to the image.

The same principle is used in a differential windlass, where the connected pulleys are replaced by winches.

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 .