You are riding a stationary exercise bike. Assume the rear wheel is twenty0eight inches in diameter,...

You are riding a stationary exercise bike. Assume the rear wheel is twenty0eight inches in diameter, the rear sprocket has a radius of two inches and the front sprocket has a radius of five inches. How many resolutions per minjute of the front sprocket produces a forward speed of forty milers per hour on the bike?

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Expert Solution

Hey there!

Hope you are safe being in these Lockdown days.Now, let me answer this question for you.. First lets draw the diagram for what the question really asks for.

With the given question I have marked the radius in each circle. First let us find the circumference of each circles here. We know that the circumference of a circle is . By substituting the value of the radius 2 , 5 and 14 (diameter is 28 inches) in each circle , we get the values of their circumference as 12.56 inches , 31.42 inches and 87.97 inches.

Now by dividing the circumference of the sprocket of radius of 5 inches by the sprocket of radius 2 inches. we will know the number of rotations that the small sprocket spins when the bigger sprocket spins for 1 time.

That is 31.42 / 12.568 = 2.5 ROTATIONS IN SMALL SPROCKET WHEN THE BIGGER SPROCKET IS ROTATED FOR 1 TIME.

But the fact is that since the small sprocket is attached to the center of the wheel the number of rotations that the small sprocket spins is going to be the same number of rotation the wheel spins. In that case the wheel also spins 2.5 times when the bigger sprocket is spun for 1 time.

In that case when the bigger sprocket rotates for 1 time, the distance by covered by the wheel is 2.5 x (circumference of the wheel) = 2.5 x 87.97 = 220 inches

Well, let me give you an information of how much inches a mile is. It is 63360 inches in a mile.

In the question it talks about 40 miles per hour , and that means the wheel has to cover 40 miles , that is 40 x 63360 inches = 2534400 inches in an hour.

We already know that with one rotation of the bigger sprocket, the wheel covers 220 inches. In that case to cover 2534400 , the bigger sprocket has to rotate

I hope you get the idea. Cheers :)

genius_generous answered 1 year ago

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