Rigid Body Rotation relative velocity question

In summary, the concept of rigid body rotation involves viewing the effect of two velocities on a rigid body as a sum of translation and rotation. The rotation component is fixed at one point and has a relative velocity of the other point with respect to it. This relative velocity vector is necessarily perpendicular to the line between the two points due to the constraint of rigidity. This is not an assumption, but a provable consequence of the assumption of rigidity.
  • #1
Master1022
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This is a question about the concepts behind rigid body rotation when we use relative velocity.

In general, let us say that we have a rigid body and on it are two points, A and B, which are moving with velocities vA and vB respectively. These velocities are in random directions.

The theory states that we can view the effect of these velocities as the sum of a translation and a rotation. I am comfortable with the above and understand it.

My question lies in the nature of the rotation component. Let us say that our translation consisted of the vA components. Thus the rotation component will be 'fixed' at point A and will have a relative velocity of B with respect to A, BvA, acting at point B. Why is the relative velocity vector (BvA) necessarily normal/perpendicular to the line between A and B?

My thoughts are the following: if we have randomly chosen the velocity vectors of A and B, how can we assume that BvA = vB - vA is perpendicular to BrA?

The explanations in the textbook are slightly contradictory in my opinion. In one case, they say that: [itex] \textbf{relative v} = \omega \times \textbf{r} [/itex], but they also seem to construct vector diagrams of velocity vectors (v_a, v_b, and b_v_a)- the latter approach makes no sense in that the direction of the relative velocity vector is set by the velocity vectors, which could be random.

Thanks for reading this and any clarification is greatly appreciated.
 
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  • #2
Master1022 said:
if we have randomly chosen the velocity vectors of A and B, how can we assume that BvA = vB - vA is perpendicular to BrA?
The rigidity of the body imposes an additional constraint on vA and vB - they must be such that the distance between points A and B does not change. That additional constraint is sufficient to justify this assumption.
(so it's not really an assumption, it's a provable consequence of the assumption of rigidity)
 
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  • #3
Nugatory said:
The rigidity of the body imposes an additional constraint on vA and vB - they must be such that the distance between points A and B does not change. That additional constraint is sufficient to justify this assumption.
(so it's not really an assumption, it's a provable consequence of the assumption of rigidity)
Thank you very much for your response- it makes sense now.
 

1. What is rigid body rotation?

Rigid body rotation is a type of motion in which an object moves in a circular path around a fixed axis without any deformation or change in shape.

2. What is relative velocity in rigid body rotation?

Relative velocity in rigid body rotation refers to the velocity of one point on the rotating object relative to another point on the same object. It takes into account both the linear and angular velocities of the object.

3. How is relative velocity calculated in rigid body rotation?

Relative velocity in rigid body rotation can be calculated using the formula v = ω x r, where v is the relative velocity, ω is the angular velocity, and r is the distance between the two points.

4. What factors affect relative velocity in rigid body rotation?

The relative velocity in rigid body rotation is affected by the angular velocity, the distance between the two points, and the direction of rotation. It is also affected by any external forces acting on the object.

5. How is rigid body rotation different from other types of motion?

Rigid body rotation is different from other types of motion, such as linear motion, because it involves both rotational and translational motion. In rigid body rotation, all points on the object move in circular paths around a fixed axis, whereas in linear motion, the object moves in a straight line.

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