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------------------------------------------------ Consider the planar 4R mechanism illustrated in the figure. For this linkage, the constant link lengths, in generic units, are:

- [tex]a_1 = 6[/tex]
- [tex]a_2 = 11[/tex]
- [tex]a_3 = 8[/tex]
- [tex]a_4 = 14[/tex]

Answer the following question and show all of your work:

(a) Determine the mobility limits for link [tex]d_1[/tex] relative to the 2D-axis using an algebraic method. In other words, state the limits on the angle [tex]\theta_1[/tex]: Is [tex]a_2[/tex] a crank, rocker, 0-rocker, or 7-rocker?

For a planar 4R mechanism, the mobility can be calculated using Gruebler's equation. The 'az' link, with a length of 11 units, is considered a crank in this mechanism due to its capability of making a full 360-degree rotation. The angular momentum of the rigid body is summed from individual mass segments.

Explanation:

In Engineering, a planar 4R mechanism refers to a four-bar linkage mechanism where all the moving parts are in a single plane. It includes four links, with one being the fixed link. In this case, the links are of lengths ai = 6, az = 11, a3 = 8, and ay = 14.

For mobility limits of links in a planar mechanism, a formula called Gruebler's equation is used. In this problem, the Gruebler's equation works out as M = 3(n-1) - 2j - h, where n is the number of links, j is the number of lower pairs, and h is the number of higher pairs. For a planar 4R mechanism, all pairs are lower pairs.

The angular momentum could be developed by computing the magnitude of angular momentum for each individual mass segment in the rigid body, and then summing them all up.

The 'az' link can be defined either as a crank, rocker, 0-rocker, or 7-rocker based on its motion. A crank is a link that makes a complete 360-degree rotation, while a rocker oscillates between two limits. Since az=11 units is capable of a complete rotation without hitting any adjacent links, it is considered a crank.

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