Let's analyze the answers and corrections based on the given diagram and provided details.

Problem Summary:

A rod of 10 cm length is attached at one end (Point A) to a point on a wheel of radius 5 cm rotating counterclockwise. Point A moves in a circular path, while Point B (the other end of the rod) moves horizontally along a line passing through the wheel's center. The wheel completes 2.0 revolutions per second.

Solutions Review:

Here are the provided answers along with my analysis:

(a) Coordinates of Point A and B at $t = \frac{1}{12}$ sec:

1. Correct answer for Point A: $\left( \frac{5}{2}, \frac{5 \sqrt{3}}{2} \right)$.
2. Incorrect answer for Point B:
   • The coordinates should reflect Point B's horizontal displacement relative to Point A's location.
   • Correct position of Point B should be: $\left( \frac{5}{2} + 5\cos(60^\circ), \frac{5\sqrt{3}}{2} + 5\sin(60^\circ) \right)$.

(b) Angular Velocity:

• Answer Provided: $\omega = 4\pi$ radians per second.
• Analysis: This is correct. Since the wheel rotates at 2 revolutions per second, the angular velocity is $\omega = 2 \times 2\pi = 4\pi$ radians per second.

(c) Coordinates of Point A as a function of $t$:

• Correct Answer:
  • $x = 5 \cos(4\pi t)$
  • $y = 5 \sin(4\pi t)$
• Explanation: These are accurate expressions for the coordinates of Point A, considering the angular velocity $\omega = 4\pi$.

(d) x-coordinate of Point B:

• Incorrect Answer: $5 \cos(4\pi t) + 10$.
• Correct Form: To express the x-coordinate of Point B, consider the full rod length and rotation.

(e) Velocity of Point B:

• Correct Answer: $-20\pi \sin(4\pi t)$ cm/sec.
• Explanation: This represents the horizontal velocity of Point B correctly.

Let me know if you'd like further breakdowns of any part of the solution or related explanations.

Follow-up Questions:

1. What is the full derivation of the x-coordinate of Point B as a function of $t$?
2. How is the y-coordinate of Point B calculated for any given time $t$?
3. Can we verify Point A’s position using trigonometric identities?
4. How does the length of the rod influence Point B's movement?
5. Why is the angular velocity given as $4\pi$, and how does this relate to the rotation period?

Quick Tip:

When dealing with rotational motion, remember that angular velocity ($\omega$) relates directly to the number of revolutions per second and is given by $\omega = 2\pi \times \text{frequency}$.