Step 1: Analyze the problem

We are tasked with calculating the angular speed of a bicycle's wheels in radians per minute and the number of revolutions per minute.

Given:

• Diameter of wheel: 20 inches → Radius, $r = 10$ inches.
• Linear speed: 15 miles/hour.
• 1 mile = 5280 feet = $5280 \times 12 = 63360$ inches.
• Relationship between angular speed ($\omega$) and linear speed:
  $\text{Linear speed} = r \cdot \omega$

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Step 2: Convert linear speed to inches/minute

The given speed is in miles/hour, so we convert it to inches/minute: $15 \, \text{mi/h} = 15 \cdot 63360 \, \text{in/h}$ $15 \cdot 63360 = 950400 \, \text{in/h}$ Since there are 60 minutes in an hour: $950400 \, \text{in/h} \div 60 = 15840 \, \text{in/min}.$ Thus, the linear speed is $15840 \, \text{in/min}$.

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Step 3: Compute the angular speed in radians/min

Using the formula $\text{Linear speed} = r \cdot \omega$, where $r = 10$ inches: $15840 = 10 \cdot \omega$ $\omega = \frac{15840}{10} = 1584 \, \text{rad/min}.$

Final angular speed:

$\boxed{\omega = 1584 \, \text{rad/min}}$

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Step 4: Compute the revolutions per minute

The angular speed ($\omega$) in radians/min can be converted to revolutions per minute ($\text{rev/min}$) using the fact that 1 revolution = $2\pi$ radians: $\text{Revolutions per minute} = \frac{\omega}{2\pi} = \frac{1584}{2\pi}.$ Approximating $\pi \approx 3.1415927$: $\text{Revolutions per minute} = \frac{1584}{2 \cdot 3.1415927} \approx \frac{1584}{6.2831854} \approx 252.172.$

Final revolutions per minute:

$\boxed{\text{Revolutions per minute} = 252.172 \, \text{rev/min}}$

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Summary:

1. Angular speed: $\omega = 1584 \, \text{rad/min}$.
2. Revolutions per minute: $252.172 \, \text{rev/min}$.

Would you like further details or clarification?

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Related Questions:

1. How do you convert between radians and revolutions?
2. How would the calculations change for different wheel diameters?
3. What is the relationship between linear and angular velocity?
4. How does this concept apply to car tires or machinery?
5. How do you derive the formula $\omega = v/r$?

Tip: Always double-check unit conversions when working with different measurement systems.