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To solve for the missing values of \(\omega_{AB}\) (angular velocity of \(AB\)) and \(v_B\) (velocity of point \(B\)) in this mechanism, we can use relative velocity analysis.

Given dataTo solve for the missing values of \(\omega_{AB}\) (angular velocity of \(AB\)) and \(v_B\) (velocity of point \(B\)) in this mechanism, we can use relative velocity analysis.

Given 
- \(L = 450 \text{ mm}\) (length of link \(AB\))
- \(\theta = 30^\circ\)
- \(\beta = 80^\circ\)
- \(v_A = 300 \text{ mm/s}\)

### Step 1: Determine \(\omega_{AB}\)

Using the velocity relationship for a point on a rotating link:
\[
v_A = \omega_{AB} \cdot L \cdot \sin(\beta)
\]

Rearrange to solve for \(\omega_{AB}\):
\[
\omega_{AB} = \frac{v_A}{L \cdot \sin(\beta)}
\]

Substitute the values:
- \(v_A = 300 \text{ mm/s}\)
- \(L = 450 \text{ mm}\)
- \(\beta = 80^\circ\)

So,
\[
\omega_{AB} = \frac{300}{450 \cdot \sin(80^\circ)}
\]

Calculating this will give us \(\omega_{AB}\).

### Step 2: Determine \(v_B\)

Using the relative velocity equation at point \(B\) on the link:
\[
v_B = \omega_{AB} \cdot L \cdot \cos(\beta)
\]

After finding \(\omega_{AB}\) from Step 1, substitute it here to find \(v_B\).

I'll compute these values now.The missing values are as follows:

- \(\omega_{AB} = 0.677 \, \text{rad/s}\)
- \(v_B = 52.90 \, \text{mm/s}\)

These can be filled in the table's yellow cells.

Would you like a more detailed explanation of each step, or do you have any questions?

Here are some related questions to deepen your understanding:
1. How would the values of \(\omega_{AB}\) and \(v_B\) change if \(v_A\) were increased?
2. What effect would a change in angle \(\beta\) have on the velocities of points \(A\) and \(B\)?
3. How does changing the length \(L\) impact \(\omega_{AB}\) and \(v_B\)?
4. Can you calculate the angular velocity if the angle \(\theta\) changes instead of \(\beta\)?
5. How would this problem change if the velocity of point \(A\) were directed at an angle?

**Tip:** For mechanisms, relative velocity analysis can be a quick way to solve for unknowns when dealing with rotational and translational components.

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 determine the angular velocity (ω_AB) and the velocity of point B (v_B) in a collar-link mechanism with given kinematic parameters. This problem utilizes relative velocity analysis, kinematics, and trigonometry concepts, making it suitable for undergraduate engineering students.

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