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A rigid body constrained at a distance $r$ from its center of mass with a ball-and-socket type constraint is considered to only have three (rotational) degrees of freedom. But isn't rigid body's center of mass still technically translating when it rotates with respect to the pivot of the constraint? This type of constrained rigid body can clearly have both linear and angular acceleration. So why then isn't it considered to still have 6 degrees of freedom?

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2 Answers 2

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Because translational and angular velocities for such a system are tied together; once you know one, you can directly tell the other. Therefore the system has only three degrees of freedom.

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  • $\begingroup$ I understand thanks! Are position and rotation of the body also tied together in this case? Can you derive one from another? $\endgroup$ Mar 7, 2020 at 15:18
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    $\begingroup$ Yes, you only need three variables to describe the position and orientation of this body in space. $\endgroup$
    – Tofi
    Mar 7, 2020 at 15:22
  • $\begingroup$ I understand. Am I correct to assume that the relationship between angular and linear velocities would be $v = w×r$? Not sure how to express the relationship between angular and linear accelerations though. $\endgroup$ Mar 7, 2020 at 15:32
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I Worte the equations for your case

enter image description here

enter image description here

Where S is the rotation matrix between body fixed system and Inertial system

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