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Figure 5 illustrates a slider-crank mechanism which has two rotational joints at points A and B and a slider at point C. When *l1* = *l2* =  0.4 metre, θB = 90o and the speed of the slider is 3m/s, determine the angular velocity of the link *l2*.

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.

Learn how to determine the angular velocity of link l2 in a slider-crank mechanism. Given l1 = l2 = 0.4 meters, θB = 90 degrees, and slider speed vC = 3 m/s. Detailed step-by-step solution included. Suitable for advanced engineering studies.

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