https://file.aiquickdraw.com/mathbot/file/1730794063556t34vi9gq.jpg

Let's analyze and solve the problem in the image step-by-step.

### Question 1 Analysis
The question provides information about a mechanism where:
- Collar A moves to the right with velocity \( v_A = 300 \, \text{mm/s} \).
- The bar \( AB \) has an angular velocity \( \omega_{AB} \) and length \( L = 450 \, \text{mm} \).
- We are given two angles, \( \theta = 30^\circ \) and \( \beta = 80^\circ \).

The task is to fill in the table by calculating:
- \( \omega_{AB} \) (angular velocity of the bar \( AB \))
- \( v_B \) (velocity of point \( B \))

### Solution Approach
To solve this, we can break down the velocity of point \( B \) (end of bar \( AB \)) into components and use kinematic relationships.

#### 1. Angular Velocity \( \omega_{AB} \)
Given that:
- The velocity at point \( B \) can be expressed in terms of \( \omega_{AB} \) as it rotates around point \( A \).
- The relation \( v_B = v_A + \omega_{AB} \times AB \) is used.

Based on the working shown in the image, the angular velocity \( \omega_{AB} \) is calculated as:
\[
\omega_{AB} = -1.02 \, \text{rad/s}
\]
This value appears to be determined from the equation involving the angle and distances provided.

#### 2. Velocity of \( B \), \( v_B \)
From the given notes:
- \( v_B \) is broken into components based on the angle \( \beta \).
- Using trigonometric components for velocities along the axes:
  - \( v_{Bx} = 0.3 + 0.45 \omega \cos(60^\circ) \)
  - \( v_{By} = 0.45 \omega \sin(60^\circ) \)

With \( \omega = -1.02 \, \text{rad/s} \), we can plug it in to find \( v_B \):
\[
v_B = 404 \, \text{mm/s}
\]
This is the resultant velocity at point \( B \) when combining the components.

### Table Completion
Now, we fill in the missing values in Table 1:

| \( L \) (mm) | \( \theta \) (deg) | \( \beta \) (deg) | \( v_A \) (mm/s) | \( \omega_{AB} \) (rad/s) | \( v_B \) (mm/s) |
|--------------|---------------------|--------------------|------------------|---------------------------|-------------------|
| 450          | 30                  | 80                | 300              | **-1.02**                 | **404**           |

---

Would you like a detailed breakdown of any specific calculation steps or further clarification?

### Further Questions
1. How is the angular velocity \( \omega_{AB} \) calculated based on the given angles and velocities?
2. Why is \( \omega_{AB} \) negative, and what does this signify about the direction of rotation?
3. How are the trigonometric components used to find \( v_B \)?
4. What are the implications if \( \beta \) or \( \theta \) angles were changed?
5. How would the calculation differ if \( v_A \) was directed to the left instead of the right?

### Tip
For kinematic problems involving rotating links, always decompose velocities into components aligned with the motion direction to simplify calculations.

Collar A moves at a velocity, v_A, to the right. The values of the kinematic parameters are given in the table. Complete the table by filling in the values in the yellow cells.

Learn how to calculate the angular velocity ω_AB and velocity v_B for a collar and bar mechanism. This problem includes relative velocity and angular velocity calculations with trigonometric components, ideal for engineering students.

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