A4-centimeter rod is attached at one end to a point *A* rotating counterclockwise on a wheel of radius2cm. The other end *B* is free to move back and forth along a horizontal bar that goes through the center of the wheel. At time *t=0* the rod is situated as in the diagram at the left below. The wheel rotates counterclockwise at2.5revolutions per second. Thus, when *t=1/15* sec, the rod is situated as in the diagram at the right below. Note that the wheel has its center at the origin.

To analyze the motion of the rod and its attachment to the rotating wheel, we can break down the problem into a few key components:

1. **Wheel Rotation**: The wheel rotates at 2.5 revolutions per second. This corresponds to an angular velocity \(\omega\) given by:
   \[
   \omega = 2.5 \times 2\pi \text{ rad/s} = 5\pi \text{ rad/s}
   \]

2. **Position of Point A**: At any time \(t\), the position of point \(A\) on the circumference of the wheel can be described using the parametric equations:
   \[
   A(t) = (2 \cos(\omega t), 2 \sin(\omega t))
   \]

3. **Rod Movement**: The rod length is 4 cm. The position of point \(B\) can vary based on the angle that the rod makes with the horizontal bar. 

At \(t = 0\) seconds, we can denote the initial position of \(A\) as:
   \[
   A(0) = (2, 0)
   \]
and point \(B\) is at \( (x_B, 0) \) on the horizontal bar.

At \(t = \frac{1}{15}\) seconds, we calculate the new position of \(A\):
   \[
   A\left(\frac{1}{15}\right) = (2 \cos\left(5\pi \cdot \frac{1}{15}\right), 2 \sin\left(5\pi \cdot \frac{1}{15}\right))
   \]

Calculating these:
   \[
   \omega t = 5\pi \cdot \frac{1}{15} = \frac{\pi}{3}
   \]
   \[
   A\left(\frac{1}{15}\right) = (2 \cos(\frac{\pi}{3}), 2 \sin(\frac{\pi}{3})) = (1, \sqrt{3})
   \]

4. **Position of B**: Given that the length of the rod is 4 cm, the position of \(B\) will depend on the angle of the rod with respect to the horizontal. As \(B\) moves along the horizontal bar, the constraint is that the distance \(AB\) equals 4 cm. The coordinates of \(B\) will need to satisfy:
   \[
   (x_B - x_A)^2 + (y_B - y_A)^2 = 4^2
   \]

If you want to delve into specifics like the position of \(B\) at \(t = \frac{1}{15}\) seconds or the subsequent motion, please clarify your focus. 

Do you have any specific questions or need details on a particular part of this analysis? Here are some related questions you might consider:
1. What is the position of point \(B\) at \(t = 0\)?
2. How does the angle of the rod affect the position of \(B\)?
3. What are the coordinates of \(B\) when \(t = \frac{1}{15}\) seconds?
4. How can we describe the motion of \(B\) over time?
5. What other factors could influence the movement of the rod? 

**Tip**: Visualize the problem by sketching the positions of \(A\) and \(B\) at different times to better understand the relationships.

Explore the dynamics of a 4-centimeter rod attached to a rotating wheel with a radius of 2 cm, spinning at 2.5 revolutions per second. This problem breaks down the movement of points A and B over time, providing key equations and positions at specific time intervals. Suitable for advanced students.

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