Let's analyze the questions in the image.

Activity 1:

1. A rider notices that the wheels of his bicycle make 12 revolutions in 15 seconds.

   • a. What is the average angular speed of the wheel in radians?
   • b. What distance in meters does the wheel travel if its radius is 33 cm?

2. What is the angular velocity of an object traveling in a circle of radius 0.75 m with a linear speed of 3.5 m/s?

1(a) Average Angular Speed:

• Angular speed (ω) is calculated by: $\omega = \frac{\theta}{t}$ where:

  • $\theta$ is the angular displacement in radians,
  • $t$ is the time in seconds.

  The wheel completes 12 revolutions, and since each revolution equals $2\pi$ radians: $\theta = 12 \times 2\pi = 24\pi \, \text{radians}$ Time, $t = 15$ seconds. Therefore, the angular speed is: $\omega = \frac{24\pi}{15} = \frac{8\pi}{5} \approx 5.03 \, \text{radians/second}.$

1(b) Distance traveled:

• The distance traveled in one complete revolution is the circumference of the wheel, which is: $\text{Circumference} = 2\pi r$ Given that the radius $r = 33 \, \text{cm} = 0.33 \, \text{m}$, the distance traveled per revolution is: $2\pi \times 0.33 = 2.07 \, \text{m}$ Since the wheel makes 12 revolutions, the total distance traveled is: $12 \times 2.07 = 24.84 \, \text{m}.$

2. Angular Velocity:

• Angular velocity (ω) can also be found using the linear velocity (v) and the radius (r) by the formula: $\omega = \frac{v}{r}$ Given that $v = 3.5 \, \text{m/s}$ and $r = 0.75 \, \text{m}$: $\omega = \frac{3.5}{0.75} \approx 4.67 \, \text{radians/second}.$

Would you like more details on any of these calculations or additional clarifications?

Here are five related questions to expand on the concepts:

1. How does angular velocity differ from linear velocity in circular motion?
2. What happens to the angular velocity if the radius of the wheel is increased but the linear speed remains the same?
3. How can you calculate the total angular displacement after a certain number of revolutions?
4. What is the relationship between frequency (in revolutions per second) and angular speed?
5. How would the total distance change if the rider took 20 seconds instead of 15 for the same number of revolutions?

Tip: Always ensure you convert units properly (e.g., cm to meters) when calculating physical quantities!