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There is a rigid body undergoing general planar motion (translation+rotation) in 2D. I dont know the translational velocity and center of mass(CM), all I know is the velocity at geometric center (which is not the CM) and rotational velocity. So if I knew either translational velocity or CM, I can find the other and calculate kinetic energy.

What I did was to discretize points on rigid body and calculate Kinetic Energies if those points were CMs.

My question is that, is there any condition/theorem constraining Kinetic Energy, so I can say for example minimum or maximum Kinetic energy values of those points are likely to be the value associated with real CM?

[PS: I need such Kinetic energy condition (if any) to combine it with my equations of motion for an optimization routine. So what I need is KE condition independent of other equations.]

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    $\begingroup$ Any part of the body, including its geometric centre, is moving in a circle around the centre of mass. How dud you obtain the translational velocity without determining the centre of mass? $\endgroup$
    – my2cts
    Jan 10 at 10:07
  • $\begingroup$ I basically took one point in the rigid body and calculated if it were the center of mass and body was rotating about this point. And since I know velocity at geometric center and I have assumed center of rotation, I can derive translational velocity. And I repeated this for a lot of points on the surface of the body. With those values I can derive kinetic energy for each scenario (different points). Now I wanted to check if there is a physical condition that would differentiate those points from each other. $\endgroup$ Jan 10 at 11:29
  • $\begingroup$ so, don't you know your mass distribution? Not even in a parametric way, since you're talking about an optimization routine. You're creating a discrete model of your system, but how do you define the mass of these points/regions? $\endgroup$
    – basics
    Jan 10 at 12:14
  • $\begingroup$ Could you also add a sketch of the system, and any other detail that could help who reads to better understand your question? $\endgroup$
    – basics
    Jan 10 at 12:15
  • $\begingroup$ look at this equation $~ \overrightarrow{v}_{p}=\overrightarrow{v}_{c}+\overrightarrow{\omega}\times \overrightarrow{r}_{cp}~$ where c is the center of mass and p is rigid body point. you know $~ v_p~,\omega~$ and you are looking for $~v_c~,r_{cp}~$ to obtain the unknowns, you need more information $\endgroup$
    – Eli
    Jan 10 at 12:39

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i dont think that is possible.. For example, if you stand on ground, the particles inside you are moving, ie. they do have kinetic energy in microscopic view. But in no way that means that you kinetic energy. (also you can't argue that kinetic energies cancel each other, as mass cant be negative and neither can v^2.) so for the idea of taking discrete points and integrating etc., that isnt possible. There may be others ways to do that, which i may not know of.

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  • $\begingroup$ What I meant is that I think there is one point in the body which represents the CM (and rotation). I dont know where it is so I cant derive translational velocity, (both are unknowns). But since I know the shape of body I have at least possible interval of values for CM. Using this, I generate many points and based on them I calculate the translational velocity, if they were actual CM. Hence, I can derive Kinetic energy of each point. My question: is there any law so I can use as criterion and distinguish these possible candidates of points and say maybe CM locates in some interval. $\endgroup$ Jan 10 at 11:44

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