For part (d), we need to determine the x-coordinate of the right end of the rod at point $B$ as a function of time $t$.

Given information:

1. The rod rotates with point $A$ on a circle of radius $4$ cm, with center at the origin, and point $A$ has a circular motion with angular velocity $\omega = 4\pi$ radians per second.
2. The rod length $AB = 8$ cm, with $B$ moving along the x-axis.

To express the x-coordinate of point $B$ in terms of $t$:

1. Position of Point $A$:

   • From part (c), the coordinates of point $A$ are: $x_A = 4 \cos(4\pi t)$ $y_A = 4 \sin(4\pi t)$

2. Determining the x-coordinate of $B$:

   • The x-coordinate of $B$ can be calculated by extending from $A$ by the length of the rod $AB = 8$ cm along the horizontal direction.
   • Since point $B$ lies horizontally along the x-axis relative to $A$, its x-coordinate is given by: $x_B = x_A + 8 \cos(4\pi t)$
   • This becomes: $x_B = 4 \cos(4\pi t) + 8 \cos(4\pi t) = (4 + 8) \cos(4\pi t) = 12 \cos(4\pi t)$

Thus, the x-coordinate of point $B$ as a function of $t$ is: $x_B = 12 \cos(4\pi t)$

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

Related Questions:

1. How would the answer change if the wheel rotated at a different speed?
2. What is the y-coordinate of point $B$, and why does it remain zero?
3. How would you calculate the velocity of point $B$ at any time $t$?
4. How can we find the angular position of point $A$ as a function of $t$?
5. What happens to point $B$’s motion if the rod length $AB$ changes?

Tip: To determine the x-coordinate of a point moving in relation to another, consider both the length and the angle of the connecting segment.