Planar Linkage with 8 Links and 9 Simple Revolute Joints

To determine the number of degrees of freedom of a planar linkage, we need to apply Gruebler's equation. Gruebler's equation states that the number of degrees of freedom (DOF) of a planar linkage is given by:

DOF = 3(n - 1) - 2j - h

Where:
- n is the number of links in the linkage
- j is the number of simple revolute joints in the linkage
- h is the number of higher pair joints in the linkage

In this case, we have a planar linkage with:
- n = 8 (8 links)
- j = 9 (9 simple revolute joints)
- h = 0 (no higher pair joints)

Applying Gruebler's Equation

Using Gruebler's equation, we can calculate the number of degrees of freedom:

DOF = 3(8 - 1) - 2(9) - 0
DOF = 3(7) - 18
DOF = 21 - 18
DOF = 3

Therefore, the number of degrees of freedom of the planar linkage with 8 links and 9 simple revolute joints is 3.

Explanation

In a planar linkage, each simple revolute joint restricts the motion along one degree of freedom. The total number of degrees of freedom is equal to the degrees of freedom contributed by the links minus the degrees of freedom restricted by the joints.

In this case, we have 8 links, which means there are 8 degrees of freedom contributed by the links. However, we have 9 simple revolute joints, which restrict the motion along 2 degrees of freedom each (one for rotation and one for translation). Since there are more joints than the degrees of freedom they restrict, we need to subtract the excess number of joints from the degrees of freedom contributed by the links.

Therefore, the total number of degrees of freedom is 3, which means the planar linkage can move independently along 3 different directions.

Conclusion

The correct answer is option C) 3. The planar linkage with 8 links and 9 simple revolute joints has 3 degrees of freedom.

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