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:

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.