Understanding Gear Ratios and Wheel Revolutions: A Practical Example

When mechanical systems like gears and wheels are involved, understanding the relationship between the revolutions of different components is crucial. This article explores how to determine the number of revolutions a smaller wheel makes when a larger wheel undergoes a specific number of revolutions. We will use the example of two wheels, each with a specific radius, to walk through the process step-by-step.

The Problem

We have a large toothed wheel with a radius of 20 cm and a smaller toothed wheel with a radius of 15 cm. If the larger wheel makes 3 full revolutions, how many revolutions does the smaller wheel make?

Calculating Circumferences

The first step in solving this problem is to calculate the circumferences of both wheels. The formula for the circumference of a circle is given by:

[ C 2 pi r ]

Larger Wheel Circumference

For the larger wheel with a radius of 20 cm:

[ C_{text{large}} 2 pi r_{text{large}} 2 pi (20) 40 pi text{ cm} ]

Smaller Wheel Circumference

For the smaller wheel with a radius of 15 cm:

[ C_{text{small}} 2 pi r_{text{small}} 2 pi (15) 30 pi text{ cm} ]

Distance Traveled by the Larger Wheel

The second step involves calculating the distance traveled by the larger wheel when it makes 3 full revolutions. The distance traveled is equal to three times the circumference of the larger wheel:

[ text{Distance}_{text{large}} 3 times C_{text{large}} 3 times 40 pi 120 pi text{ cm} ]

Calculating the Revolutions of the Smaller Wheel

To find out how many revolutions the smaller wheel makes, we divide the distance traveled by the larger wheel by the circumference of the smaller wheel:

[ text{Revolutions}_{text{small}} frac{text{Distance}_{text{large}}}{C_{text{small}}} frac{120 pi}{30 pi} 4 ]

Thus, the smaller wheel makes 4 revolutions when the larger wheel makes 3 revolutions.

Additional Insights

The concept presented here is a fundamental principle in the field of mechanical engineering and is known as gear ratio. The gear ratio is the ratio of the number of revolutions of the driving gear (larger wheel) to the number of revolutions of the driven gear (smaller wheel).

Conclusion

Understanding how the number of revolutions of different gears or wheels relate to each other is crucial in various engineering and scientific applications. Whether you're working with gears, pulleys, or simple wheels, the relationship between their sizes and the number of revolutions can be calculated using the circumference and the distance traveled.

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