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 Math Emeritus Sci Advisor Thanks PF Gold P: 39,568 How do gears provide a mechanical advantage? There is a law of "conservation of energy", not "conservation of force" and all "mechanical advantage" laws are based on that. If you have a cog of radius R m and turn it through on complete turn, $2\pi$ radians, by applying force F Newtons, then its surface has moved through a distance of $2\pi R$ m and so you have done $2\pi RF$ Joules work on it. If a chain or other ratcheting mechanism causes another cog, of radius r, to turn through the same distance, by "conservation of energy" you have done $2\pi r f= 2\pi RF$ Joules of work on it also and so must have applied $f= (2\pi RF)/(2\pi r)= (R/r)F$ Newtons force. "R/r" is the "mechanical advantage".