Question

a bicycle with 24 - inch diameter wheels is traveling at 11 mi/h. find the angular speed of the wheels in radians per minute. 968.000 radians per minute. how many revolutions per minute do the wheels make? (round your answer to three decimal places.) revolutions per minute

Explanation:

Step1: Convert speed from miles - per - hour to inches - per - minute

1 mile = 5280 feet and 1 foot = 12 inches, so 1 mile = 5280×12 = 63360 inches.
11 mi/h = 11×63360 inches/hour.
To convert from inches - per - hour to inches - per - minute, divide by 60.
11×63360÷60 = 11584 inches per minute.

Step2: Find the circumference of the wheel

The formula for the circumference of a circle is \(C=\pi d\), where \(d = 24\) inches. So \(C = 24\pi\) inches.

Step3: Calculate the number of revolutions per minute

The number of revolutions per minute \(n=\frac{\text{linear speed}}{\text{circumference}}\).
\(n=\frac{11584}{24\pi}\approx153.938\) revolutions per minute.

Step4: Calculate the angular speed in radians per minute

One revolution is \(2\pi\) radians.
The angular speed \(\omega\) (in radians per minute) is \(\omega = 2\pi n\).
Since \(n=\frac{11584}{24\pi}\), then \(\omega=2\pi\times\frac{11584}{24\pi}=\frac{11584}{12}\approx965.333\) radians per minute.

Answer:

The number of revolutions per minute is approximately 153.938.
The angular speed is approximately 965.333 radians per minute.