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In the mechanism shown here, points C, A and D are in same straight line. If CA = 20 mm, CD = 40 mm and ω2 = 4 rad/s (CCW), then the value of ω4 in rad/s is:

In the mechanism shown here, points C, A and D are in same straight line. If CA = 20 mm, CD = 40 mm and ω2 = 4 rad/s (CCW), then the value of ω4 in rad/s is: Correct Answer 4.0 (CW)

Concept:

Kennedy  theorem:

For the relative motion between the number of links in the mechanism, for any of the 3 links, the three instantaneous centers must lie in a straight line.

Here, we know the angular velocity of link 2 and we have to find it for link 4. So, First, find the  common I-center for link 2 and 4 that means, I24

When 4 links are there, then total number of I- centers will be = 4C2 = 6

Links:     1              2              3              4             

12           13           14          

23           24

34

Here according to the Kennedy theorem,  I24 I21,  and I14  will lie on the same straight line and I24 will be the intersection of lines joining I21, I14, and lines joining I23, I34.

About I24, links 2 and 4 will be under pure rotation.

Angular velocity Theorem: When the angular velocity of a link is known (ω2) and it is required to find the angular velocity of another link (ω4), locate their common I - center (I24). The velocity of this I - center relative to a fixed third link (1) is the same whether the I - center is considered on the first or on the second link.

⇒ VA = ω2 (I24 I12) = ω4 (I24 I14)         ....(1)

Calculation: 

Given:

ω2 = 4 rad/s (CCW), CD = 40 mm, CA = 20 mm, ω4 = ?

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By using equation (1),

ω2 (I24 I12) = ω4 (I24 I14)

ω2 (CA) = ω4 (CD - CA)

4 (20) = ω4 (20)

ω4 = 4 rad/s (CW)

The direction of ω4 and ω2 are just opposite because I12 and I14 lie on the opposite side of I24.

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