Ticktock Town has a clock which long resides in the memory of all who visit there. As shown above, the clock consists of 12 small identical circular plates, numbered and seated on a circle so that each touches both its neighbours. The clock also has a wheel that runs around the outside of the clockface, has the same radius as each of the small plates, and has its centre attached by a rod and spring to the minute hand. The spring keeps tension on the wheel in the direction of the clock centre so that as the minute hand rotates the wheel rolls along the outer contours of the 12 plates without losing surface contact. How many times does the wheel rotate with respect to the clock centre each hour?
The wheel rotates six times each hour. Let the circumference of each small plate beC. As shown, the wheel runs over C/4 of each plate. There are two additivecomponents of the wheel rotation. If the circumference of a plate were stretched out into a linear path, since the radii of wheel and plate are equal, the wheel rotates C/4 due to the path length. Also the tangent to the plate at the point of wheel contact rotates through 90 degrees as the wheel rolls over the plate so an extra C/4 rotation of the wheel occurs for each plate. This gives a total wheel rotation of 6C for 12 plates
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| The wheel rotates six times each hour. Let the circumference of each small plate beC. As shown, the wheel runs over C/4 of each plate. There are two additivecomponents of the wheel rotation. | |
| If the circumference of a plate were stretched out into a linear path, since the radii of wheel and plate are equal, the wheel rotates C/4 due to the path length. Also the tangent to the plate at the point of wheel contact rotates through 90 degrees as the wheel rolls over the plate so an extra C/4 rotation of the wheel occurs for each plate. This gives a total wheel rotation of 6C for 12 plates | ||
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