Let's solve the problem in the image step by step.

Step 1: Given Information

• Diameter of the wheels: $24$ inches.
• Speed of the bicycle: $14 \, \text{mi/h}$.

We need to find:

1. The angular speed $\omega$ in radians per minute.
2. The number of revolutions per minute (rpm).

Step 2: Convert Linear Speed to Angular Speed

Relationship Between Linear Speed and Angular Speed: $v = \omega r$ where:

• $v$ is the linear speed of the wheel (in inches per minute).
• $\omega$ is the angular speed in radians per minute.
• $r$ is the radius of the wheel.

First, convert the given speed from miles per hour to inches per minute:

1 mile = 63,360 inches, and 1 hour = 60 minutes.

So: $14 \, \text{mi/h} = 14 \times 63,360 \, \text{inches/hour} = 887,040 \, \text{inches/hour}$ Now convert to inches per minute: $v = \frac{887,040}{60} = 14,784 \, \text{inches/minute}$

Find the radius of the wheel:

The diameter is 24 inches, so the radius $r$ is: $r = \frac{24}{2} = 12 \, \text{inches}$

Calculate the angular speed:

Using $v = \omega r$: $\omega = \frac{v}{r} = \frac{14,784}{12} = 1,232 \, \text{radians/minute}$

Step 3: Calculate Revolutions per Minute

1 revolution is $2\pi$ radians. To convert the angular speed to revolutions per minute: $\text{Revolutions per minute} = \frac{\omega}{2\pi} = \frac{1,232}{2\pi}$ $\text{Revolutions per minute} \approx \frac{1,232}{6.2832} \approx 196.11 \, \text{revolutions per minute}$

Final Answers:

• Angular speed $\omega$: $1,232 \, \text{radians/minute}$.
• Revolutions per minute: $196.111 \, \text{rpm}$ (rounded to three decimal places).

Would you like further clarification or details on any of the steps?

Here are 5 related questions to deepen your understanding:

1. How does the angular speed change if the diameter of the wheels increases?
2. What would happen to the revolutions per minute if the bike's speed were doubled?
3. Can you convert the angular speed to degrees per second?
4. How would the calculations differ if the wheel's radius was given in meters instead of inches?
5. What is the linear speed in meters per second if the wheel diameter is 0.61 meters?

Tip: Always keep track of unit conversions, especially when switching between different systems like miles to inches or radians to revolutions.