Application Problem - Bicycle Gear

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• Jan 31st 2009, 08:04 PM
chrozer
Application Problem - Bicycle Gear
A bicycle's gear ratio is the number of times the freewheel turns for every one turn of the chainweel (see figure). The table shows the number of teeth in the freewheel and chainwhee for the first five gears of an 18-speed touring bicycle. The chainwheel completes one rotation for each gear. Find the angle through which the freewheel turns for each gear.

I have attached a picture with the data of the problem.

I have no idea where to start with this application problem. I just finished learning about the Unit Circle and almost everything about Trigonometry, but I can't figure this out. Help?
• Feb 1st 2009, 01:13 AM
earboth
Quote:

Originally Posted by chrozer
A bicycle's gear ratio is the number of times the freewheel turns for every one turn of the chainweel (see figure). The table shows the number of teeth in the freewheel and chainwhee for the first five gears of an 18-speed touring bicycle. The chainwheel completes one rotation for each gear. Find the angle through which the freewheel turns for each gear.

I have attached a picture with the data of the problem.

I have no idea where to start with this application problem. I just finished learning about the Unit Circle and almost everything about Trigonometry, but I can't figure this out. Help?

You only have to calculate how many revolutions the freewheel (r) performs when the chainwheel has done a complete rotation.

First gear: $\displaystyle r = \dfrac{teeth\ of\ chainwheel}{teeth\ of\ freewheel}$
That means:

$\displaystyle r = \dfrac{24}{32} = \dfrac34$

If you use the first gear the freewheel performs $\displaystyle \frac34$ revolutions if the chainwheel has completed one revolution.

One revolution correspond to $\displaystyle 2\pi$ or 360°. Therefore the freewheel performs an angle of $\displaystyle \frac34 \cdot 2\pi=\frac32 \pi$ or $\displaystyle \frac34 \cdot 360^\circ = 270^\circ$

All other calculations have to be done similarly.
• Feb 1st 2009, 07:34 AM
chrozer
Shouldn't the ratio be inverted? Shoulnd't it be $\displaystyle r = \dfrac{teeth\ of\ freewheel}{teeth\ of\ chainwheel}$?
• Feb 1st 2009, 07:47 AM
earboth
Quote:

Originally Posted by chrozer
Shouldn't the ratio be inverted? Shoulnd't it be $\displaystyle r = \dfrac{teeth\ of\ freewheel}{teeth\ of\ chainwheel}$?

No. The smaller the freewheel is the more revolution it performs if the chainwheel rotates exactly one time.

Actually the ratio tells you how often the number of teeth of the chainwheel contains the number of the freewheel.
• Feb 1st 2009, 09:00 AM
chrozer
Ok thanx alot.