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.