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.