https://file.aiquickdraw.com/mathbot/file/1730352031177e71i7u9x.jpg

Let's solve each part of the problem.

### Given:
- Rod length = 8 cm
- Wheel radius = 4 cm
- Wheel rotation = 2.0 revolutions per second, counterclockwise
- Initial time \( t = 0 \) is the left position shown in the diagram.

Since the wheel rotates at \(2.0\) revolutions per second, we can convert this into radians per second, as each revolution is \(2\pi\) radians.

### Solution

#### Part (a): Coordinates of Points A and B at \( t = \frac{1}{12} \) seconds

1. **Angular Velocity Calculation**:
   \[
   \omega = 2.0 \times 2\pi = 4\pi \text{ radians per second}
   \]

2. **Angular Position of Point A**:
   The angle \(\theta\) of point A at \( t = \frac{1}{12} \) seconds can be calculated using:
   \[
   \theta = \omega \cdot t = 4\pi \cdot \frac{1}{12} = \frac{\pi}{3} \text{ radians}
   \]

3. **Coordinates of Point A**:
   Since the center of the wheel is at the origin and the radius is \(4\) cm, the coordinates of point \( A \) (located on the circle's circumference) are:
   \[
   x_A = 4 \cos\left(\frac{\pi}{3}\right) = 4 \times \frac{1}{2} = 2
   \]
   \[
   y_A = 4 \sin\left(\frac{\pi}{3}\right) = 4 \times \frac{\sqrt{3}}{2} = 2\sqrt{3}
   \]
   So, \( A \) is at \( (2, 2\sqrt{3}) \).

4. **Coordinates of Point B**:
   Since \( B \) is \(8\) cm away from \(A\) along the rod in a horizontal direction, the x-coordinate of \( B \) would be:
   \[
   x_B = x_A + 8 = 2 + 8 = 10
   \]
   \[
   y_B = 0
   \]
   So, \( B \) is at \( (10, 0) \).

### Part (b): Angular Velocity \(\omega\)
As calculated above,
\[
\omega = 4\pi \text{ radians per second}
\]

### Part (c): Parametric Equations for Point A
To express the x and y coordinates of point \( A \) as a function of time \( t \), we use the general parametric form for circular motion:
\[
x = R \cos(\omega t)
\]
\[
y = R \sin(\omega t)
\]
Substituting \( R = 4 \) cm and \( \omega = 4\pi \):
\[
x = 4 \cos(4\pi t)
\]
\[
y = 4 \sin(4\pi t)
\]

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### Final Answers

- **(a)** Coordinates at \( t = \frac{1}{12} \) sec:
  - \( A = (2, 2\sqrt{3}) \)
  - \( B = (10, 0) \)
- **(b)** Angular velocity \(\omega = 4\pi\) radians per second
- **(c)** Parametric equations for \( A \):
  - \( x = 4 \cos(4\pi t) \)
  - \( y = 4 \sin(4\pi t) \)

Would you like further clarification or details on any part of this solution?

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### Related Questions
1. How would the coordinates of \( B \) change if the rod length was different?
2. What would be the coordinates of \( A \) and \( B \) after one full rotation?
3. How does changing the wheel's rotational speed affect the angular velocity?
4. How can you calculate the instantaneous velocity of point \( A \) at any given time?
5. What would happen to the motion if the rod were rigidly fixed rather than sliding?

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### Tip
Always remember that in circular motion, angular velocity \(\omega\) is directly proportional to the rotational speed. More rotations per second mean a higher angular velocity.

A rod of 8 cm is attached at one end to a point A rotating counterclockwise on a wheel of radius 4 cm. The other end, B, is free to move along a horizontal bar through the center of the wheel. When t = 1/12 sec, find the coordinates of points A and B, the angular velocity of point A, and express the coordinates of point A as a function of time t.

Explore how to calculate the coordinates of points A and B on a rotating rod attached to a wheel, given an 8 cm rod length, a wheel radius of 4 cm, and a 2 revolutions per second rotation rate. Learn about angular velocity, circular motion, and trigonometric functions.

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