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Let's analyze the answers and corrections based on the given diagram and provided details.

### Problem Summary:
A rod of 10 cm length is attached at one end (Point A) to a point on a wheel of radius 5 cm rotating counterclockwise. Point A moves in a circular path, while Point B (the other end of the rod) moves horizontally along a line passing through the wheel's center. The wheel completes 2.0 revolutions per second.

### Solutions Review:
Here are the provided answers along with my analysis:

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#### (a) Coordinates of Point A and B at \( t = \frac{1}{12} \) sec:
1. **Correct answer for Point A:** \( \left( \frac{5}{2}, \frac{5 \sqrt{3}}{2} \right) \).
2. **Incorrect answer for Point B:** 
   - The coordinates should reflect Point B's horizontal displacement relative to Point A's location.
   - Correct position of Point B should be: \( \left( \frac{5}{2} + 5\cos(60^\circ), \frac{5\sqrt{3}}{2} + 5\sin(60^\circ) \right) \).
   
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#### (b) Angular Velocity:
- **Answer Provided**: \( \omega = 4\pi \) radians per second.
- **Analysis**: This is correct. Since the wheel rotates at 2 revolutions per second, the angular velocity is \( \omega = 2 \times 2\pi = 4\pi \) radians per second.

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#### (c) Coordinates of Point A as a function of \( t \):
- **Correct Answer**: 
   - \( x = 5 \cos(4\pi t) \)
   - \( y = 5 \sin(4\pi t) \)
- **Explanation**: These are accurate expressions for the coordinates of Point A, considering the angular velocity \( \omega = 4\pi \).

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#### (d) x-coordinate of Point B:
- **Incorrect Answer**: \( 5 \cos(4\pi t) + 10 \).
- **Correct Form**: To express the x-coordinate of Point B, consider the full rod length and rotation.

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#### (e) Velocity of Point B:
- **Correct Answer**: \( -20\pi \sin(4\pi t) \) cm/sec.
- **Explanation**: This represents the horizontal velocity of Point B correctly.

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Let me know if you'd like further breakdowns of any part of the solution or related explanations. 

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### Follow-up Questions:

1. What is the full derivation of the x-coordinate of Point B as a function of \( t \)?
2. How is the y-coordinate of Point B calculated for any given time \( t \)?
3. Can we verify Point A’s position using trigonometric identities?
4. How does the length of the rod influence Point B's movement?
5. Why is the angular velocity given as \( 4\pi \), and how does this relate to the rotation period?

#### Quick Tip:
When dealing with rotational motion, remember that angular velocity (\( \omega \)) relates directly to the number of revolutions per second and is given by \( \omega = 2\pi \times \text{frequency} \).

A 10-centimeter rod is attached at one end to a point A rotating counterclockwise on a wheel of radius 5 cm. The other end B is free to move along a horizontal line. The wheel rotates at 2.0 revolutions per second. Determine the coordinates, angular velocity, and velocity of points A and B as functions of time t.

Analyze a rod rotating counterclockwise on a wheel and determine the coordinates and velocities of its endpoints. This problem covers concepts in circular motion, angular velocity, and trigonometric transformations suitable for Grades 11-12 or early college.

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