The document contains a series of mechanical engineering problems focusing on the analysis of slider-crank mechanisms and linkages, primarily involving velocity analysis. Below is an outline of the key questions from the document:

1. Slider-Crank Mechanism (Velocity Analysis):

   • Given $AB = l$, $BC = \sqrt{7}l$, with crank AB rotating at a constant angular velocity $\omega$, the task is to find the velocity of the slider at different crank angles ($\theta = 60^\circ, 90^\circ, 120^\circ, 180^\circ$).

2. Velocity of Linkage Points:

   • For a different slider-crank mechanism, with the slider moving at 1 m/s, the question asks for the velocity and angular velocity of point B and crank AB, respectively.

3. Four-Bar Linkage:

   • For a four-bar linkage with constant angular velocity $\omega$ for link AB, the problem requires finding the velocity of point C and the angular velocities of links BC and CD.

4. Instantaneous Center of Rotation:

   • Another four-bar linkage problem with a square plate attached between BC asks to locate the instantaneous center and find the angular velocities around it and for points on the plate.

5. Offset Cylinder Mechanism:

   • The problem involves an offset slider-crank mechanism, where the crank rotates at angular velocity $\omega$, and the goal is to determine the slider's velocity, stroke, and other related quantities.

These problems require using kinematic principles such as velocity analysis through geometry, angular velocity, and link length relations.

If you'd like detailed solutions for any specific problem from this set, please let me know!

Here are five follow-up questions to expand on these topics:

1. How do you derive the velocity of a slider in a slider-crank mechanism from its geometric parameters?
2. What is the role of the instantaneous center of rotation in analyzing four-bar linkages?
3. How can angular velocity affect the velocity of other links in a linkage system?
4. How is the stroke of a slider calculated in an offset slider-crank mechanism?
5. How does the position of a crank (e.g., $\theta$) influence the velocity of connected components?

Tip: When solving these problems, it's useful to apply vector diagrams or velocity polygons to visualize the relationships between angular and linear velocities.