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.