To find the coordinates of Point $B$ at $t = \frac{1}{12}$ seconds, we need to analyze the rotation of Point $A$ around the origin.

Given Information:

1. The wheel has a radius of $4$ cm.
2. Point $A$ is attached to a rod of length $8$ cm, with $B$ being free to slide horizontally.
3. The wheel rotates at $2.0$ revolutions per second counterclockwise.
4. At $t = 0$, the rod is horizontal, with $A$ located at $(4, 0)$.

Step-by-Step Solution:

1. Calculate the angle at $t = \frac{1}{12}$ sec:

   • The wheel completes $2.0$ revolutions per second, meaning each revolution takes $\frac{1}{2}$ seconds.
   • In $\frac{1}{12}$ seconds, the angle rotated is: $\text{Angle} = 2 \cdot 2\pi \cdot \frac{1}{12} = \frac{\pi}{3} \text{ radians}$

2. Determine coordinates of Point $A$:

   • Since the radius of the wheel is $4$ cm, and the wheel rotates counterclockwise, the new position of $A$ forms an angle of $\frac{\pi}{3}$ with the positive $x$-axis.
   • Therefore, the coordinates of $A$ at $t = \frac{1}{12}$ sec are: $x = 4 \cos\left(\frac{\pi}{3}\right) = 4 \cdot \frac{1}{2} = 2$ $y = 4 \sin\left(\frac{\pi}{3}\right) = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3}$
   • So, $A$ is at $(2, 2\sqrt{3})$.

3. Determine coordinates of Point $B$:

   • Point $B$ lies horizontally to the right of $A$, at a distance of $8$ cm.
   • Since $A$ is at $(2, 2\sqrt{3})$, $B$'s $x$-coordinate will be $2 + 8 = 10$, and its $y$-coordinate will be the same as $A$'s $y$-coordinate, which is $0$.

Final Answer:

• Point $A$ coordinates: $\left(2, 2\sqrt{3}\right)$
• Point $B$ coordinates: $(10, 0)$

Would you like further details on this solution or have any questions?

Related Questions

1. How would the coordinates change if the wheel rotated clockwise?
2. What would happen if the rod length was different, say $10$ cm?
3. How would the coordinates of $B$ vary if $B$ could slide vertically instead of horizontally?
4. What is the angular velocity of Point $A$ in radians per second?
5. How does the period of rotation affect the positions of $A$ and $B$?

Tip

Remember that angular displacement in radians can be found by multiplying the angular velocity by time.