Sprocket Ratio Calculations

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The easiest way to calculate sprocket ratio is to count the number of teeth on both the driving and the driven sprockets and divide the first by the second. This ratio tells you how many times the driven sprocket turns for every revolution of the driving sprocket. From this, you can calculate revolutions per minute (rpm) for the driven sprocket. This is the same way you calculate gear ratio for a gear chain. Unlike a gear train, a sprocket train uses a chain, but the chain does not enter into the calculation of sprocket ratio.

Driving and Driven Sprockets

Motorcycles and bicycles have two sprockets. The one attached to the pedals or to the engine crankshaft is the driving sprocket, and the one connected to the rear wheel is the driven sprocket. The driving sprocket is almost always larger than the driven one, and as you shift gears upwards, the chain engages with progressively larger driving sprockets in the front while simultaneously shifting to smaller ones in the rear. This increases the sprocket ratio, which makes it harder to pedal while increasing the rotational speed of the rear wheel. Motorcycle sprockets work in essentially the same way, except that it's the engine that has to work harder in higher gears, not the rider.

Calculating the Sprocket Ratio

The sprocket ratio is a function of the relative sizes of the driving and driven sprockets, and while you could calculate it by dividing their diameters, it's easier to just count teeth. The sprocket ratio is simply the number of teeth on the driving sprocket (T1) divided by the number of teeth on the driven sprocket (T2).

  • Sprocket ratio = T1/T2

If the front sprocket on a bicycle has 20 teeth and the rear sprocket has 80, the sprocket ratio is 20/80 = 1/4 = 1:4 or simply 4.

Relative Revolutions Per Minute

A larger sprocket ratio may make a bicycle more difficult to pedal, but that's because it increases the rotational speed of the rear wheel, and that makes the bicycle go faster. On the other hand, a small socket ratio makes it easier to accelerate. The ratio of the rotational speed of the driven sprocket (V2) in rpms relative to those of the driving sprocket (V1) is the same as the sprocket ratio.

  • Sprocket ratio = T1/T2 = V1/V2

If you're pedaling a bicycle with a sprocket ratio of 4 – which is typically the practical maximum – and you turn the driving sprocket at a speed of 60 rpm, the rear sprocket and the rear wheel spin at:

  • 1/4 = 60/V2 rpm; V2 = 240 rpm   

Calculating Vehicle Speed

Knowing the rotational velocity of the rear wheel, you can calculate the forward speed of the vehicle if you know the wheel diameter. After measuring it, multiply it by π to get the circumference of the wheel. Assuming no slippage, the vehicle moves forward by this amount with every revolution. Multiply that by the number of revolutions per minute, and you have the forward velocity. If you measure the wheel in inches, your answer is in inches per minute, and you may want to convert that to miles per hour to get a more meaningful number.

A Sample Calculation

You can use this information to calculate the speed of a bicycle with a 28-inch rear wheel and a maximum gear ratio of 3.5 when the rider is able to turn the pedals at a speed of 40 rpm. The radius of the rear wheel is (28/2) = 14 inches, so its circumference is 2π(14) = 87.92 inches. That's how far the bicycle travels with each revolution of the wheel.

The rider is turning the pedals at 40 rpm and the gear ratio is 3.5, so the rear wheel is rotating at 140 rpm. That means that, in one minute, the bicycle travels a distance of 12,309 inches. A speed of 12,309 inches/ minute is equivalent to 0.194 miles/minute, which equals 11.64 miles/hour.

References

About the Author

Chris Deziel holds a Bachelor's degree in physics and a Master's degree in Humanities, He has taught science, math and English at the university level, both in his native Canada and in Japan. He began writing online in 2010, offering information in scientific, cultural and practical topics. His writing covers science, math and home improvement and design, as well as religion and the oriental healing arts.

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