The circular disc shown in its plan view in the figure rotates in a plane parallel to the horizontal plane about the point $$O$$ at a uniform angular velocity $$\omega $$. Two other points $$A$$ and $$B$$ are located on the line $$OZ$$ at distances $${r_A}$$ and $${r_B}$$ from $$O$$ respectively.

The velocity of point $$B$$ with respect to point $$A$$ is a vector of magnitude

The lengths of the links of a 4-bar linkage with revolute pairs only are $$p,q,r$$ and $$s$$ units. Given that $$p < q < r < s.$$ Which of these links should be the fixed one, for obtaining a ‘double crank’ mechanism?

The mechanism used in a shaping machine is

The overall gear ratio in a 2 stage speed reduction gear box (with all spur gears) is 12. The input and output shafts of the gear box are collinear. The countershafts which is parallel to the input and output shafts has a gear (Z₂ teeth) and pinion (Z₃ = 15 teeth) to mesh with pinion (Z₁ = 16 teeth) on the input shafts and gear (Z₄ teeth) on the output shafts respectively. It was decided to use a gear ratio of 4 with 3 module in the first stage and 4 module in the second stage.

Z₂ and Z₄ are