How fast must the leftmost 36 tooth gear rotate so that the 84 tooth gear will rotate once per second?

To determine how fast the leftmost 36-tooth gear must rotate for the 84-tooth gear to rotate once per second, we can use the principle of gear ratios.

The gear ratio can be calculated by taking the number of teeth on the output gear (84-tooth) and dividing it by the number of teeth on the input gear (36-tooth). This gives us:

[ \text{Gear Ratio} = \frac{\text{Teeth on 84-tooth gear}}{\text{Teeth on 36-tooth gear}} = \frac{84}{36} = \frac{7}{3} ]

This ratio indicates that for every 7 rotations of the 36-tooth gear, the 84-tooth gear will complete 3 rotations. To find out how many rotations the 36-tooth gear needs to make for the 84-tooth gear to complete 1 rotation, we can rearrange the ratio:

[ \text{Rotations of 36-tooth gear} = \frac{3}{7} \text{ rotations of the 84-tooth gear} ]

Since we want the 84-tooth gear to rotate once per second:

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