Let's solve each part of the problem.

Given:

• Rod length = 8 cm
• Wheel radius = 4 cm
• Wheel rotation = 2.0 revolutions per second, counterclockwise
• Initial time $t = 0$ is the left position shown in the diagram.

Since the wheel rotates at $2.0$ revolutions per second, we can convert this into radians per second, as each revolution is $2\pi$ radians.

Solution

Part (a): Coordinates of Points A and B at $t = \frac{1}{12}$ seconds

1. Angular Velocity Calculation: $\omega = 2.0 \times 2\pi = 4\pi \text{ radians per second}$

2. Angular Position of Point A: The angle $\theta$ of point A at $t = \frac{1}{12}$ seconds can be calculated using: $\theta = \omega \cdot t = 4\pi \cdot \frac{1}{12} = \frac{\pi}{3} \text{ radians}$

3. Coordinates of Point A: Since the center of the wheel is at the origin and the radius is $4$ cm, the coordinates of point $A$ (located on the circle's circumference) are: $x_A = 4 \cos\left(\frac{\pi}{3}\right) = 4 \times \frac{1}{2} = 2$ $y_A = 4 \sin\left(\frac{\pi}{3}\right) = 4 \times \frac{\sqrt{3}}{2} = 2\sqrt{3}$ So, $A$ is at $(2, 2\sqrt{3})$.

4. Coordinates of Point B: Since $B$ is $8$ cm away from $A$ along the rod in a horizontal direction, the x-coordinate of $B$ would be: $x_B = x_A + 8 = 2 + 8 = 10$ $y_B = 0$ So, $B$ is at $(10, 0)$.

Part (b): Angular Velocity $\omega$

As calculated above, $\omega = 4\pi \text{ radians per second}$

Part (c): Parametric Equations for Point A

To express the x and y coordinates of point $A$ as a function of time $t$, we use the general parametric form for circular motion: $x = R \cos(\omega t)$ $y = R \sin(\omega t)$ Substituting $R = 4$ cm and $\omega = 4\pi$: $x = 4 \cos(4\pi t)$ $y = 4 \sin(4\pi t)$

Final Answers

• (a) Coordinates at $t = \frac{1}{12}$ sec:
  • $A = (2, 2\sqrt{3})$
  • $B = (10, 0)$
• (b) Angular velocity $\omega = 4\pi$ radians per second
• (c) Parametric equations for $A$:
  • $x = 4 \cos(4\pi t)$
  • $y = 4 \sin(4\pi t)$

Would you like further clarification or details on any part of this solution?

Related Questions

1. How would the coordinates of $B$ change if the rod length was different?
2. What would be the coordinates of $A$ and $B$ after one full rotation?
3. How does changing the wheel's rotational speed affect the angular velocity?
4. How can you calculate the instantaneous velocity of point $A$ at any given time?
5. What would happen to the motion if the rod were rigidly fixed rather than sliding?

Tip

Always remember that in circular motion, angular velocity $\omega$ is directly proportional to the rotational speed. More rotations per second mean a higher angular velocity.