Sunday, November 04, 2007

Text Savvy: Odometer Errors and Tire Sizes Part One

Text Savvy
An odometer records the number of kilometers a car travels. Assume that the odometer gives the correct distance when the overall diameter of the wheel plus tire is 50 cm. If several months later, 6.8 mm of the tread was worn evenly from the tire, what percentage more or less than the actual distance traveled will the odometer reading give?
Tires: my bane in Vermont. I’ve blogged about them before. I deliberately know as little as possible about cars and I had to do a bit of research in order to solve this problem. I found this informative pdf file by Brian Burke that helped me understand the relationships involved with tire size, odometer errors and speed. I also found “Affect of tire size on odometer readings” by George Mastros. I followed it word for word in order to solve the problem. Looking back, I know that if I had understood odometers, I would have quickly solved it myself because this is a familiar circumference problem. To atone for this, I then figured out what the odometer error is when I have my winter studded tires on. I also roughly determined the wear on my tires and created a tongue-in-cheek Doggie Miles calculator.

If the diameter of the tire is 50 cm, then circumference is 50π, or 157.1 cm or 1.57 m. The worn tire lost 6.8 mm on both sides of each tire. 6.8 cm is a loss of 13.6 mm in all, which is 1.36 cm. The circumference of the worn tire is, therefore, 48.64π, or 1.53 m.

One revolution of each tire equals the circumference of the tire. I need to calculate the revolutions required to travel one kilometer:

New tire: 1000 m ÷ 1.57 m = 636.94 revolutions/km
Worn tire: 1000 m ÷ 1.53 m = 653.59 revolutions/km

In other words, for every 636.94 revolutions of the new tire, the odometer will record 1 kilometer.

At this point I get confused again because of my conflicting sources (and because of my continuing confusion about odometers):

Every time the wheel turns 636.94 revolutions, the worn tires will travel only 974.52 m (revolutions • circumference of worn tire: 636.94 revolutions • 1.53 m). The worn tire will have only gone 974.52 meters — or 25.48 meters short of a kilometer. The odometer will “click” 1 kilometer every 636.94 revolutions. So the odometer will read higher than the car has traveled.

How much will the error be? 1000 m ÷ 25.48 m = 39.25 km. Therefore, the odometer will click one extra kilometer every 39.25 km. The odometer will actually read 40.25 km. The odometer error: 974.52 m.

However, Burke says that the odometer error calculation is (new tire diameter ÷ old tire diameter): 0.4864 m ÷ 0.5 m = 0.9728 m. According to this calculation, for every 1000 kilometers clicked on the odometer, the car has traveled 972.8 km.

This is a small error between the two methods but I cannot account for why it exists. If you can gently explain it to me in comments, I would appreciate it.

I will continue this post tomorrow, when I will do the same calculations with my snow tires.

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3 comments:

  1. I haven't been around lately trying to catch up.

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  2. 1 old rev = 50πcm
    1 new rev = 48.64πcm (but reads 50π).

    New tire goes distance D in D/50π revolutions. Odometer reads D/50π

    Worn tire goes distance D in D/48.6π revolutions. Odometer reads D/50π.

    Worn tire goes 1000m in in 1000m/.486πm = 654.959 revolutions.

    Odometer reads 654.959*.5π = 1028.807m.

    When the car travels 1km, the odometer reads 1.0289. Taking the reciprocal, when the odometer reads 1km, the car will have traveled 0.972km.

    I could have found these percent errors by dividing 50 by 48.6, and 48.6 by 50.

    So what about your different answer?

    Don't laugh. You used 3.14 for π, while Burke and I used the π key on the calculator. And you rounded twice, the first time when you calculated the circumferences.

    Your 636.94 is my 636.62, and your 653.59 is my 654.96.

    I think the rest is good.

    ReplyDelete
  3. Hi Peppy! I haven't been around lately, either!

    Thank you so much Jonathan: it's such a simple thing, those rounding errors: I never even considered it, but I should have because I know how rounding π can give them.

    ReplyDelete

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