The problem presented involves a slider-crank mechanism, where we need to determine the angular velocity of the link $l_2$. Given:

• $l_1 = l_2 = 0.4$ meters
• $\theta_B = 90^\circ$
• The speed of the slider at point C, $v_C = 3$ m/s

Step 1: Analyze the system geometry

For the slider-crank mechanism:

• $l_1$ is the length of the crank (from A to B).
• $l_2$ is the length of the connecting rod (from B to C).
• $\theta_B = 90^\circ$ is the angle at point B, which means the crank is perpendicular to the horizontal axis.

Step 2: Establish the velocity relationship

Using the kinematic equations for the slider-crank mechanism, the velocity of the slider (point C) can be related to the angular velocities of the links. The velocity of the slider $v_C$ is given by:

$v_C = l_2 \cdot \omega_{l2} \cdot \sin(\theta_C)$

Where:

• $\omega_{l2}$ is the angular velocity of link $l_2$.
• $\theta_C$ is the angle between the connecting rod $l_2$ and the horizontal axis.

Step 3: Consider the position of the crank at $\theta_B = 90^\circ$

At $\theta_B = 90^\circ$, the connecting rod $l_2$ is also perpendicular to the slider's motion, which means $\theta_C$ would be 0 or 180 degrees (assuming the horizontal direction). Therefore, $\sin(\theta_C) = 1$ or $\sin(\theta_C) = -1$.

Step 4: Solve for the angular velocity $\omega_{l2}$

Given that $v_C = 3$ m/s:

$v_C = l_2 \cdot \omega_{l2}$

$3 = 0.4 \cdot \omega_{l2}$

Solving for $\omega_{l2}$:

$\omega_{l2} = \frac{3}{0.4} = 7.5 \text{ rad/s}$

Final Answer

The angular velocity of the link $l_2$ is $7.5$ rad/s.

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Would you like more details on any specific part of the solution, or do you have any other questions?

Related Questions:

1. How would the angular velocity change if the speed of the slider increased?
2. What happens to the angular velocity if the length of $l_2$ is doubled?
3. How do you derive the velocity relationship in a slider-crank mechanism?
4. What is the effect of changing the crank angle $\theta_B$ on the angular velocity?
5. How do you calculate the angular acceleration of the link $l_2$?
6. What are the forces acting on the slider-crank mechanism at point C?
7. How does the position of point B affect the overall motion of the system?
8. What other methods can be used to determine the angular velocity in such mechanisms?

Tip:

When analyzing kinematic mechanisms, always start with the geometric relationships and then apply the kinematic equations systematically. This approach helps in understanding how the different components interact.