Answer to April 14, 2003 Problemfrom The Math Forum |
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The Gear Problem Benjamin Banneker, an 18th century mathematician who lived on his parents' Patapsco River farm in Baltimore County, was the first person ever to build a wooden clock in the United States. By taking apart a pocket watch, Banneker figured out that the ratio of the gears was the key to making it all work. Banneker's clock struck every hour for more than forty years, keeping perfect time.Suppose you have these three gears:
Think of them meshing like this:
If Gear C moves clockwise three revolutions,
Will Gear B move clockwise or counter-clockwise?
Will Gear A move clockwise or counter-clockwise? |
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Solution to Problem: If C made 3 revolutions, then:gear B made 3 * (12 / 6) = 6 revolutions in a counter clockwise direction; gear A made 3 * (12 / 3) = 12 revolutions in a clockwise direction.
First, you must find how many teeth each gear has. To work, gears must go different directons. That means that gear B will go counter clockwise and gear A will go clockwise. Compare the number of teeth in gears C and B. Gear C had 12 teeth, gear B had 6. To find the number of revolutions, you divide the number of teeth in C by the number of teeth in B. 12 / 6 = 2. So, gear B makes 2 revolutions for each revolution that gear C makes. Hence, gear B will make 6 revolutions for the three that gear C makes. Now compare the number of teeth in gears B and A. Gear B had 6 teeth, but it has to be multiplied by 2 because it is going to revolve 2 times. A had 3 teeth. To find the number of revolutions, you divide the number of teeth in B * 2 by the number of teeth in A. So, gear A will make 4 revolutions for each revolution that gear C makes; therefore, when gear C makes three revolutions, gear A will make 4 revolutions.
(# of revolutions are shown in the table)
Gear C Gear B Gear A
-------- -------- --------
1 2 4
2 4 8
3 6 12
I. If C made one revolution, then:
- B made 1 * (12 / 6) = 2 revolutions;
- A made 1 * (12 / 3) = 4 revolutions.
II. If C made 2 revolutions, then:
- B made 2 * (12 / 6) = 4 revolutions;
- A made 2 * (12 / 3) = 8 revolutions.
III. If C made n revolutions, then:
- B made n * (12 / 6) = 2n revolutions;
- A made n * (12 / 3) = 4n revolutions.
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