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Nuts & Bolts: Math Is Hard

Posted in How To: Tech Qa on April 27, 2018
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I have a 1993 Ford Bronco with a 5.0L and 33-inch mud-terrains. If I remove an accessory with a pulley outside diameter of 5.35 inches and replace it with an accessory with a pulley o.d. of 4.5 inches, is there a mathematical way to determine how much shorter a serpentine belt I would need? The alternative is to go buy eight different belts with varying lengths. I'd like to avoid that if possible.


There are two possible ways to attack the problem. The first is to figure out the difference in the circumference of the two pulleys, which involves multiplying the diameter of the circle by pi (3.14) and then subtracting, like so: 16.77 minus 14.13 is 2.64 inches. But this is total circumference, and the belt doesn’t go all the way around the pulley; you only need to concern yourself with the portion where the belt actually touches the pulley. You didn’t mention how much engagement the belt had around the pulley, but let’s say it’s about half of it. In this case, half of 2.64 inches is 1.32, so you’d need a belt that is about 1.5 inches shorter than your current one.

A lower-tech way to figure out how long a belt you need is to do your pulley swap and then grab some insulated wire of relatively thin gauge, like 18-22 gauge. We like using wire because it’s flexible but has less stretch than a piece of rope or string. Route the wire as you would the belt, stretch it tight, and mark where the ends meet. Remove your wire, measure it, and you have your belt length.

A good parts guy will be able to look up belts by length, and most of the time belt part numbers incorporate length and width somewhere in the part number. Just make sure you select a belt that matches your existing belt’s width and number of ribs, as serpentine belt widths and rib counts can vary. We have used this method several times, and if we don’t get it in the first shot we are usually within half an inch of the length we need.

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