How do you find the net Mechanical Advantage (MA) of two joint machines. Do you add or multiply the individual MA?

Suppose I have two sets of wheel and axle connected by a fixed pulley. Each of the wheel has a radius of 100cm and each of the axle has a radius of 10cm. What will be their combined MA? I am quite sure that both of them has a MA of 10. But what will be their total MA? Is it 20 or 100? Should I add them or multiply them?

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It is multiplication.

When you turn the wheel at the beginning by $x$ degrees, the axle at the end will turn by $x/100$ degrees. This comparison of lengths/angles of movement directly translates to the mechanical advantage. The work is a constant, so because of $W=x_1 F_1=x_2 F_2=x_1/100 F_2=x_1/100\cdot 100 F_1$, it follows that $F_2=100 F_1$, which is exactly the mechanical advantage.

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Suppose you crank the first wheel for a metre. It will move a rope attached to it's axle by 0.1m, hence the mechanical advantage is 10. Now suppose the rope from the first wheel is connected to the second wheel so it moves the second wheel 0.1m. The second wheel will move a rope attached to is axle by 0.01m. You cranked the first wheel for a metre and moved the second rope by 0.01m, so the mechanical advantage is 100. In this case you multiply the two separate mechanical advantages.

However I would be cautious about applying this rule generally as professors are quite capable of setting exam questions where it doesn't apply. The only reliable method is to go through the system and work out what the final movement is.

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