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Let’s go through the problem step by step. Here’s an analysis and solution for each part of the problem:

### Given Information:
- A rod of length 8 cm is attached at one end (Point \( A \)) to a wheel of radius 4 cm.
- The wheel rotates counterclockwise at \( 2.0 \) revolutions per second (rps), and its center is at the origin.
- The other end of the rod (Point \( B \)) can move along a horizontal line.

The time \( t = 0 \) represents the initial position of the rod, and at \( t = \frac{1}{12} \) sec, the rod is shown rotated to a new position.

### (a) Coordinates of Points \( A \) and \( B \) at \( t = \frac{1}{12} \) sec
From the solution provided:
- Point \( A \) at \( t = \frac{1}{12} \) sec has coordinates \( \left( 2, 2\sqrt{3} \right) \).
- Point \( B \) has coordinates \( \left( 2 + 2\sqrt{13}, 0 \right) \).

These are correct based on the given rotations and lengths.

### (b) Angular Velocity \( \omega \) of Point \( A \)
Since the wheel rotates at \( 2.0 \) rps:
1. Convert revolutions per second to radians per second:
   \[
   \omega = 2 \times 2\pi = 4\pi \text{ radians per second}
   \]

### (c) Coordinates of Point \( A \) as a Function of \( t \)
The coordinates of \( A \) on the wheel will follow a circular path, so we can express its position as:
\[
x_A = 4 \cos(4\pi t)
\]
\[
y_A = 4 \sin(4\pi t)
\]

### (d) X-Coordinate of Point \( B \) as a Function of \( t \)
Since \( B \) is at a fixed distance of 8 cm from \( A \), its position along the x-axis will depend on the rotation of \( A \):
1. Find the x-coordinate by projecting point \( B \) along the line that is 8 cm away from \( A \):
   \[
   x_B = x_A + 8 \cos(4\pi t) = 4 \cos(4\pi t) + 8 \cos(4\pi t)
   \]

Would you like further details on any step?

### Follow-up Questions:
1. How would the solution change if the wheel rotated clockwise?
2. How does increasing the radius of the wheel affect the coordinates of points \( A \) and \( B \)?
3. What if point \( B \) could move vertically? How would this affect the function?
4. What would be the angular velocity if the wheel rotated at 3 revolutions per second?
5. Could you express the coordinates of point \( B \) in polar form?

### Tip:
Always convert rotational speed to radians per second when dealing with angular velocity in circular motion problems.

A rod of 8 cm is attached at one end to a rotating wheel of radius 4 cm. The wheel rotates counterclockwise at 2 revolutions per second, and the other end of the rod moves along a horizontal bar. Given the setup, calculate various properties such as coordinates of specific points at t = 1/12 sec, angular velocity, and functions for coordinates of points as a function of time.

Explore the mechanics of a rotating rod attached to a wheel moving counterclockwise at 2 revolutions per second. This problem involves calculating coordinates of points, angular velocity, and time-based position functions, suitable for high school physics and mathematics students.

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