Editing Mechanical advantage (section)

== 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.<ref>Ned Pelger, [http://www.constructionknowledge.net/general_technical_knowledge/general_tech_basic_six_simple_machines.php ConstructionKnowledge.net ]</ref>

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.
[[Image:Tackles.png|thumb|360px|right|The mechanical advantage of a block and tackle equals the number of sections of rope that support the moving block; shown here it is 2, 3, 4, 5, and 6, respectively.]]

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
:<math> L = 2R + S + K, \!</math>
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 ''V''<sub>A</sub> and ''V''<sub>B</sub> of the points ''A'' and ''B'' are related by the constant length of the rope, that is
:<math>\dot{L}=2\dot{R} + \dot{S}=0,</math>
or
:<math>  \dot{S} = -2\dot{R}.</math>
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 ''V''<sub>A</sub> be positive downwards and ''V''<sub>B</sub> be positive upwards, so this relationship can be written as the speed ratio
:<math> \frac{V_A}{V_B} = \frac{\dot{S}}{-\dot{R}} = 2,</math>
where 2 is the number of rope sections supporting the moving block.

Let ''F''<sub>A</sub> be the input force applied at ''A'' the end of the rope, and let ''F''<sub>B</sub> be the force at ''B'' on the moving block.  Like the velocities ''F''<sub>A</sub> is directed downwards and ''F''<sub>B</sub> 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 ''F''<sub>A</sub>''V''<sub>A</sub> must equal the power out acting on the load ''F''<sub>B</sub>''V''<sub>B</sub>, that is
:<math>F_A V_A = F_B V_B.\!</math>

The ratio of the output force to the input force is the mechanical advantage of an ideal gun tackle system,
:<math>\mathit{MA} = \frac{F_B}{F_A} =  \frac{V_A}{V_B}  = 2.\!</math>

This analysis generalizes to an ideal block and tackle with a moving block supported by ''n'' rope sections,
:<math>\mathit{MA} = \frac{F_B}{F_A} =  \frac{V_A}{V_B}  = n.\!</math>

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.