We are given a scenario where a rod is attached to a wheel, and we are asked to express the x-coordinate of the point B (the free end of the rod) as a function of time $t$.

Step-by-Step Breakdown:

1. Wheel Rotation and Point A's Coordinates:

   • The wheel rotates counterclockwise at 3.0 revolutions per second.
   • The radius of the wheel is 2 cm.
   • At time $t = 0$, the rod is along the x-axis, meaning the point A is at the coordinates $(2, 0)$ (since the wheel's radius is 2 cm).
   • At time $t = \frac{1}{18}$ sec, we are told that point A has coordinates $(1, \sqrt{3})$.

2. Rod and Rotation:

   • The rod has a length of 4 cm, with point A fixed on the wheel.
   • The other end, point B, is free to move along the horizontal bar through the center of the wheel. The distance between A and B is constant (4 cm).

3. Angular Velocity of the Wheel:
   The angular velocity $\omega$ of the wheel is related to the number of revolutions per second: $\omega = 2\pi \times \text{(revolutions per second)} = 2\pi \times 3 = 6\pi \, \text{radians per second}.$ The angle $\theta(t)$ at time $t$ is given by: $\theta(t) = \omega t = 6\pi t.$

4. Coordinates of Point A:
   Point A is located on the wheel, and its position depends on the angle $\theta(t)$. The x- and y-coordinates of point A are: $A_x(t) = 2 \cos(6\pi t), \quad A_y(t) = 2 \sin(6\pi t).$

5. Position of Point B:

   • The distance between points A and B is 4 cm, and the rod is always aligned at an angle $\theta(t)$ with respect to the horizontal axis (since it's attached to the rotating wheel).
   • We are told that at $t = \frac{1}{18}$ sec, the coordinates of point B are $(1 + \sqrt{13}, 0)$, which gives us the information to work with the rod's orientation and length.

   From this, we can derive the position of point B relative to point A. The x-coordinate of point B is influenced by the angle of rotation and the length of the rod. Thus, the x-coordinate of point B at any time $t$ is given by the relation: $B_x(t) = A_x(t) + 4 \cos(\theta(t)).$

6. Final Expression:
   Substituting the expression for $A_x(t) = 2 \cos(6\pi t)$, we get: $B_x(t) = 2 \cos(6\pi t) + 4 \cos(6\pi t).$ Simplifying: $B_x(t) = 6 \cos(6\pi t).$

Conclusion:

The x-coordinate of point B as a function of time $t$ is: $B_x(t) = 6 \cos(6\pi t).$

Would you like further clarification on any of the steps or additional details on how this relates to other parts of the problem?

Related Questions:

1. How does the angular velocity of the wheel affect the speed of point A's motion?
2. How would the equation change if the wheel were rotating clockwise instead of counterclockwise?
3. What is the y-coordinate of point B as a function of time?
4. How can we generalize this approach for systems with multiple rotating components?
5. What would happen if the rod's length were shorter or longer than 4 cm?

Tip:

When dealing with rotational motion, always keep in mind that the position of points on the rotating object depends on both the radius (distance from the center) and the angle of rotation, which varies over time.