# Block and tackle

A block and tackle is a system of two or more pulleys with a rope or cable threaded between them, usually used to lift or pull heavy loads.

The pulleys are assembled together to form blocks and then blocks are paired so that one is fixed and one moves with the load. The rope is threaded, or reeved, through the pulleys to provide mechanical advantage that amplifies that force applied to the rope.

Hero of Alexandria described cranes formed from assemblies of pulleys in the first century. Illustrated versions of Hero's "book on raising heavy weights" show early block and tackle systems.

## Overview

File:Tackles.png
Various ways of rigging a tackle.

A block is a set of pulleys or "sheaves" mounted on a single axle. The assembly of two blocks with a rope threaded through the pulleys is called tackle. A block and tackle system amplifies the tension force in the rope to lift heavy loads. They are common on boats and sailing ships, where tasks are often performed manually.

In the diagram on the right, the number of rope sections of the tackles shown is as follows:

• Gun Tackle: 2
• Luff Tackle: 3
• Double Tackle: 4
• Gyn Tackle: 5
• Threefold Purchase: 6

Note that the Gun tackle, Double tackle and Threefold purchase all have the same number of pulleys in both blocks (one, two and three, respectively) whereas the Luff tackle and Gyn tackle have mis-matched blocks with differing numbers of pulleys.

A block and tackle is characterized by the use of a single continuous rope to transmit a tension force around one or more pulleys to lift or move a load. Its mechanical advantage is the number of parts of the rope that act on the load. The mechanical advantage of a tackle dictates how much easier it is to haul or lift the load.

If frictional losses are neglected, the mechanical advantage of a block and tackle is equal to the number of parts in the line that either attach to or run through the moving blocks -- in other words, the number of supporting rope sections.

File:Polispasto2B.jpg
A gun tackle has a single pulley in both the fixed and moving blocks with 2 rope parts (n) supporting the load (FB) of 100N. The mechanical advantage is 2, requiring a force of only 50N to lift the load.

An ideal block and tackle with a moving block supported by n rope sections has the mechanical advantage (MA),

$MA = \frac{F_B}{F_A} = n,\!$

where FA is the hauling, or input, force and FB is the load.

Consider the set of pulleys that form the moving block and the parts of the rope that support this block. If there are n of these parts of the rope supporting the load FB, then a force balance on the moving block shows that the tension in each of the parts of the rope must be FB/n. This means the input force on the rope is FA=FB/n. Thus, the block and tackle reduces the input force by the factor n.

File:Polispasto4.jpg
A double tackle has two pulleys in both the fixed and moving blocks with four rope parts (n) supporting the load (FB) of 100N. The mechanical advantage is 4, requiring a force of only 25N to lift the load.

Ideal mechanical advantage correlates directly with velocity ratio. The velocity ratio of a tackle refers to the relative velocities of the hauling line to the hauled load. A line with a mechanical advantage of 4 has a velocity ratio of 4:1. In other words, to raise a load at 1 metre per second, the hauling part of the rope must be pulled at 4 metres per second. Therefore the mechanical advantage of a double tackle is 4.

The mechanical advantage of any tackle can be increased by interchanging the fixed and moving blocks so the rope is attached to the moving block and the rope is pulled in the direction of the lifted load. In this case the block and tackle is said to be "rove to advantage."

• "Rove to advantage" – where the pull on the rope is in the same direction as that in which the load is to be moved. The hauling part is pulled from the moving block.
• "Rove to disadvantage" – where the pull on the rope is in the opposite direction to that in which the load is to be moved. The hauling part is pulled from the fixed block.

Diagram 3 shows that now three rope parts support the load W which means the tension in the rope is W/3. Thus, the mechanical advantage is three.

By adding a pulley to the fixed block of a gun tackle the direction of the pulling force is reversed though the mechanical advantage remains the same, Diagram 3a. This is an example of the Luff tackle.

While roving to advantage is the most efficient use of equipment and resources--Roving to disadvantage simply adds an extra sheave to change the direction of the pulling line. This does not change the velocity ratio but increases friction losses--there are several situations in which roving to disadvantage may be more desirable, for example when lifting from a fixed point overhead. The decision of which to use depends on pragmatic considerations for the total ergonomics of working with a particular situation.

### Friction

The formula used to find the effort required to raise a given weight is:

$F_a =\frac{L}{N} \frac{1}{\textit{eff}}$

where $F_a$ is the force applied to the hauling part of the line (the input force), $L$ is the weight of the load (the output force), $N$ is the ideal mechanical advantage of the system (which is the same as the number of segments of line extending from the moving block), and $eff$ is the mechanical efficiency of the system (equal to one for an ideal frictionless system; a fraction less than one for real-world systems with energy losses due to friction and other causes). If $S$ is the number of sheaves in the purchase, and there is a roughly $x$% loss of efficiency at each sheave due to friction, then:

$\frac{1}{\textit{eff}} \approx 1 + S \frac{x}{100}.$

This approximation is more accurate for smaller values of $S$ and $x$. A more precise estimate of efficiency is possible by use of the sheave friction factor, $K$ (which may be obtainable from the manufacturer or published tables). The relevant equation is:

$\textit{eff} = \frac{K^N-1}{K^S N (K-1)}.$

Typical $K$ values are 1.04 for roller bearing sheaves and 1.09 for plain bearing sheaves (with wire rope).

The increased force produced by a tackle is offset by both the increased length of rope needed and the friction in the system. In order to raise a block and tackle with a mechanical advantage of 6 a distance of 1 metre, it is necessary to pull 6 metres of rope through the blocks. Frictional losses also mean there is a practical point at which the benefit of adding a further sheave is offset by the incremental increase in friction which would require additional force to be applied in order to lift the load. Too much friction may result in the tackle not allowing the load to be released easily, or by the reduction in force needed to move the load being judged insufficient because undue friction has to be overcome as well.