# Why is the mechanical advantage 8 and not 4 in this pulley system?

Can someone please explain to me how the mechanical advantage in this pulley system is 8? I thought that the mechanical advantage was calculated by counting the number of strings attached to the weight, so shouldn't the MA be 4?

• Count the bits of rope that get shorter. – Andrew Steane Jul 20 at 10:14
• @AndrewSteane Awesome, simple advice (I was struggling to put in words what I did intuitively). And the reason is that the work exerted is the product of distance and force; you need to consider the moving parts because you need the distance. – Peter - Reinstate Monica Jul 21 at 10:13

The tension in the string, $$F_{\rm E}$$, is constant so the free body diagrams are as follows.

$$F_{\rm L}=8\times F_{\rm E}$$, so the mechanical advantage of the system is $$8$$.

Also note the the upward force $$F_{\rm A} = 9 \times F_{\rm E}$$.

• “The tension in the string...is constant” – that's of course where you assume friction can be neglected. In practice, friction becomes ever more of an issue as more pulleys are added. With cheap pulleys that only use simple bushings, more that three is hardly useful. To get higher advantages without low-friction pulleys, one needs to cascade multiple blocks, rather than routing a single string through all the pulleys. – leftaroundabout Jul 21 at 12:58

If you pull down on the free end with tension $$T$$, the each of the eight strings pulls up on the weight with tension $$T$$ and so the total lifting force is $$F_L=8T$$. Similarly when each of the 8 strings is shortened by a length $$l$$, the then you have $$8l$$ of extra string at the free end, so $$F_E$$ is lowered by $$8l$$. In each case the advantge comes out at $$8$$.

The arrangement shown in the picture is known as a snatch block. The mechanical advantage of a snatch block is equal to the number of supporting strands emanating from the moveable pulleys, which in your picture, are numbered "1" through "n". Since each moveable pulley has two supporting strands emanating from it, the mechanical advantage of the given system is 2n, not 8. This occurs because, assuming the same tension in each supporting strand, there are 2n tension forces pulling the load up. For more info, see https://en.wikipedia.org/wiki/Block_and_tackle

• A snatch block is a single block which opens on the side to allow the line to be "dropped in" rather than run through from one end. This is just an 8:1 arrangement of 8 simple blocks. – Matt Jul 21 at 14:52

The number of strings attached to the weight is not a measure of MA. You could double up the strings and have eight strings attached to the weight without changing the MA. Or you could join strings and have only two or even one string attached to the weight - again, the MA would not change.

Go back to first principles and work out what force is exerted on the weight when a given force $$F_E$$ is applied to the left hand rope.