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I saw this post about proving the ICAOR and this post proving the existence of it but I don't get the construction of ICAOR geometrically as shown in page-10 of this pdf.

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This centre of rotation can be reconstructed if (a) we know the directions of velocities of two points and these directions are not parallel - it is where perpendicular lines drawn from these points intersect; (b) we know the velocities of two points, and the vectors are parallel and perpendicular to the line connecting these points - we find the intersection of the line connecting the points and the line connecting the tips of velocity vectors (see the figure)

I would appreciate answers explaining how the construction comes about from the definition.

Qmechanic
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tryst with freedom
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  • Keep in mind that if a solid object is rotating about an axle, the velocity of each point in the object is tangential. – R.W. Bird Apr 02 '21 at 14:20

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It is quite similar to drawing perpendiculars to tangents of a circle in order to find its centre.
ICAOR is an axis about which the body can be considered to be rotating at a given instant.
From a geometrical point of view, this definition refers to a straight line about which particles on the rigid body are performing circular motion, where planes of these circles are perpendicular to a line i.e axis of rotation (ICAOR in this case).
So, it is clear that this line is perpendicular to plane of velocities. Therefore by considering any two points on body whose velocities are known, we can obtain ICAOR by drawing perpendiculars to these velocities and joining them. A line passing through their intersection point and perpendicular to plane of motion gives us the ICAOR.

For the case joining tips of vectors, attached is its proof: ed

Jaswanth Naga
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  • Ok this is fine but how do you explain joining lines connecting tip of vectors? (for parallel velocity vector construction) – tryst with freedom Apr 02 '21 at 17:39
  • It is a special case. Since for parallel vectors, perpendiculars coincide, we need to join tips of these vectors as an extra step. However, this is not required in case of non-parallel vectors. – Jaswanth Naga Apr 02 '21 at 18:22
  • Yes but how does connecting tips of the vectors relate to the tangent of circle idea – tryst with freedom Apr 02 '21 at 19:21
  • After connecting tips, you get two simlar triangles with a common vertex. Using their properties, you can show that the two points have same angular velocity about this vertex, which makes this common vertex as ICAOR. – Jaswanth Naga Apr 03 '21 at 02:53