https://r2.erweima.ai/mathbot/image/1726870154828qh8wu6be.jpg

The image contains two physics problems related to rotational motion and circular dynamics. I'll extract and solve both:

### Problem 1: Ferris Wheel (Rueda de la fortuna)
- **Given Data:**
  - Radius of the wheel \( r = 9 \, \text{m} \)
  - Time for one complete revolution \( T = 50 \, \text{s} \)
  
  **Find:**
  1. Angular velocity \( \omega \) (in rad/s)
  2. Tangential velocity \( v \) (in m/s)
  
  Using the formulas:
  \[
  \omega = \frac{2\pi}{T}
  \]
  \[
  v = \omega r
  \]

### Problem 2: Motorcycle Tire
- **Given Data:**
  - Constant speed of the motorcycle \( v = 13 \, \text{m/s} \)
  - Diameter of the tire \( D = 65 \, \text{cm} = 0.65 \, \text{m} \)

  **Find:**
  1. Frequency of rotation \( f \) (in Hz)
  2. Angular velocity \( \omega \) (in rad/s)

  Using the formulas:
  \[
  v = 2\pi r f
  \]
  \[
  \omega = 2\pi f
  \]

Let me now calculate these values.Here are the solutions for both problems:

### Problem 1: Ferris Wheel
- **Angular velocity \( \omega \)** = 0.13 rad/s
- **Tangential velocity \( v \)** = 1.13 m/s

### Problem 2: Motorcycle Tire
- **Frequency of rotation \( f \)** = 6.37 Hz
- **Angular velocity \( \omega \)** = 40.00 rad/s

Let me know if you'd like a detailed explanation of any part, or if you have further questions!

Here are 5 related questions for deeper exploration:
1. How would the angular velocity of the Ferris wheel change if the time for one complete revolution doubled?
2. What is the relationship between linear and angular velocity for circular motion?
3. How would the tangential velocity of the Ferris wheel change if the radius increased by 50%?
4. Why does the frequency of rotation decrease with an increase in tire diameter for a given linear velocity?
5. How would you calculate the centripetal acceleration of a point on the Ferris wheel?

**Tip:** In rotational motion, angular velocity is constant for all points on a rigid body, but tangential velocity increases with the radius.

From a Ferris wheel, 20 cabins hang at a point located exactly 9 m from the center. One cabin takes 50 seconds for a complete revolution. Find the angular and tangential velocities. Additionally, a motorcycle advances at a constant speed of 13 m/s. Calculate the rotational frequency and angular velocity of a 65 cm diameter tire.

Learn how to calculate angular and tangential velocities for a Ferris wheel, and find the rotational frequency and angular velocity for a motorcycle tire. These problems cover concepts in circular motion, angular dynamics, and frequency, suitable for students in Grades 9-12.

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