To find the rider's angular speed in revolutions per hour, we can follow these steps:

Step 1: Relate linear speed and angular speed

The relationship between linear speed $(v)$ and angular speed $(\omega)$ is given by the formula:

$v = \omega \times r$ where:

• $v$ is the linear speed (in km/hr),
• $\omega$ is the angular speed in radians per hour,
• $r$ is the radius of the wheel in kilometers.

Step 2: Calculate the radius of the wheel

The diameter of the wheel is given as 68 cm. The radius is half of that:

$r = \frac{68 \, \text{cm}}{2} = 34 \, \text{cm}$

Convert this into kilometers:

$r = \frac{34}{100000} = 0.00034 \, \text{km}$

Step 3: Solve for angular speed in radians per hour

Now, using the equation $v = \omega \times r$:

$39.532 = \omega \times 0.00034$

Solving for $\omega$:

$\omega = \frac{39.532}{0.00034} \approx 116273.53 \, \text{radians per hour}$

Step 4: Convert angular speed to revolutions per hour

Since 1 revolution is $2\pi$ radians, we can convert radians per hour to revolutions per hour by dividing by $2\pi$:

$\text{Revolutions per hour} = \frac{116273.53}{2\pi} \approx \frac{116273.53}{6.28318} \approx 18507 \, \text{revolutions per hour}$

Final Answer:

The rider's angular speed is approximately 18,507 revolutions per hour.

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Would you like more details or have any questions on this problem?

Here are 5 related questions to explore:

1. How would the angular speed change if the diameter of the wheel increased?
2. What is the angular speed in revolutions per second?
3. If the linear speed of the rider decreased to 30 km/h, what would be the new angular speed?
4. How can we convert revolutions per hour to radians per second?
5. How many total revolutions does the wheel make in 2 hours at this speed?

Tip: When converting between linear and angular motion, always ensure the units are consistent—especially for radius and speed!