Math CentralQuandaries & Queries


Question Catrina:

A car is moving at a rate of 75 miles per hour, and the diameter of its wheels is 2.6 in. Find the angular speed of the wheels in radians per minute.

Hi Catrina,

2.6 inches for the diameter of a car wheel seems unreasonable. I think it should be 2.6 feet.

There are 5,280 feet in a mile and 60 minutes in an hour so going 75 miles in an hour is equivalent to going $75 \times 5,280$ feet in 60 minutes. hence the speed of the car is

\[\frac{75 \times 5,280}{60} = 25 \times 264 \mbox{ feet per minute.}\]

The circumference of a circle is $\pi$ times the diameter so the circumference of the wheel is $\pi \times 2.6$ feet. Hence the car moves $\pi \times 2.6$ feet in one revolution. Thus the speed of the wheel is

\[\frac{75 \times 5,280}{\pi \times 2.6}\]

and the units are

\[\frac{\mbox{ feet / minute }}{\mbox{ feet / revolution }} = \frac{\mbox{ revolutions }}{ \mbox{ minute}}.\]

Thus the speed of the wheel is

\[\frac{75 \times 5,280}{\pi \times 2.6} \frac{\mbox{ revolutions }}{ \mbox{ minute}}.\]

How many radians are there in one revolution?


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