Leverage - Block and Tackle

Block and Tackle

A block and tackle is an assembly of a rope and pulleys that is used to lift loads. A number of pulleys are assembled together to form the blocks, one that is fixed and one that moves with the load. The rope is threaded through the pulleys to provide mechanical advantage that amplifies that force applied to the rope.

In order to determine the mechanical advantage of a block and tackle system consider the simple case of a gun tackle, which has a single mounted, or fixed, pulley and a single movable pulley. The rope is threaded around the fixed block and falls down to the moving block where it is threaded around the pulley and brought back up to be knotted to the fixed block.

Let S be the distance from the axle of the fixed block to the end of the rope, which is A where the input force is applied. Let R be the distance from the axle of the fixed block to the axle of the moving block, which is B where the load is applied.

The total length of the rope L can be written as

where K is the constant length of rope that passes over the pulleys and does not change as the block and tackle moves.

The velocities VA and VB of the points A and B are related by the constant length of the rope, that is

or

The negative sign shows that the velocity of the load is opposite to the velocity of the applied force, which means as we pull down on the rope the load moves up.

Let VA be positive downwards and VB be positive upwards, so this relationship can be written as the speed ratio

where 2 is the number of rope sections supporting the moving block.

Let FA be the input force applied at A the end of the rope, and let FB be the force at B on the moving block. Like the velocities FA is directed downwards and FB is directed upwards.

For an ideal block and tackle system there is no friction in the pulleys and no deflection or wear in the rope, which means the power input by the applied force FAVA must equal the power out acting on the load FBVB, that is

The ratio of the output force to the input force is the mechanical advantage of an ideal gun tackle system,

This analysis generalizes to an ideal block and tackle with a moving block supported by n rope sections,

This shows that the force exerted by an ideal block and tackle is n times the input force, where n is the number of sections of rope that support the moving block.

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