# Kinetic energy of rigid body with unknown center of mass

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.]

• 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? Jan 10 at 10:07
• 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. Jan 10 at 11:29
• 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? Jan 10 at 12:14
• Could you also add a sketch of the system, and any other detail that could help who reads to better understand your question? Jan 10 at 12:15
• 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
– Eli
Jan 10 at 12:39