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$$K_{\mathrm {translational}}= \frac{1}{2} Mv_{\mathrm {com}}^2$$

Why does the term for translational kinetic energy include only the velocity of the centre of mass of a rigid body? How can we ignore the velocity of the different particles constituting the system?

Can someone prove this to me? I tried finding it on net but couldn't as at most places there was derivation of translational $K$ for ideal gases given.

Archer
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    Isn't this just a matter of definition? By labelling $K$ as translational you are specifying that it is only the energy of the COM. The energy of motion of the components of your system would be counted as internal energy or heat. – John Rennie Sep 25 '17 at 16:28
  • But @JohnRennie the individual particles have velocity too...Also specifying $\equiv$ defining. – Archer Sep 25 '17 at 16:29
  • Yes, and we call that internal energy or heat. So total energy is the kinetic energy plus the internal energy. It isn't that we ignore the energy of the individual particles, just that we call it something else. – John Rennie Sep 25 '17 at 16:29
  • What would be the formula for that @JohnRennie ? And I strongly feel that the formula should include velocity. – Archer Sep 25 '17 at 16:30
  • We should take this to the chat – John Rennie Sep 25 '17 at 16:30
  • Purely as an aside, you will experience less confusion if you lay hands on a real mechanics book and read the introductory chapters. The treatments you find in the fat tome PHYS 101 books are incomplete because they are teaching process and point of view as much as actual physics and are trying to avoid introducing more confusion to students who aren't ready to encompass the whole subject in one big lump. But once you have the process and point of view the full treatment is less confusion. – dmckee --- ex-moderator kitten Sep 25 '17 at 17:39
  • @dmckee Real mechanics book? Can you suggest one? I am currently using Principles of Physics by Resnick, Halliday and Walker. – Archer Sep 26 '17 at 02:53
  • That is an Intro. Physics tome (tm), and not one I care for, though opinions vary rather a lot. I mean a text on classical mechanics. I used Marion and Thornton in college, but the authors are dead and the text is out of press. See https://physics.stackexchange.com/questions/9165/which-mechanics-book-is-the-best-for-beginner-in-math-major. – dmckee --- ex-moderator kitten Sep 26 '17 at 03:07
  • @dmckee I am a high school student. – Archer Sep 26 '17 at 03:11
  • That's no impediment to understanding this: the level of math you need is quite modest. Though, you need to be able to apply it with patience, persistence, and precision. – dmckee --- ex-moderator kitten Sep 26 '17 at 03:15

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The notion of 'translational' and 'rotational' kinetic energy comes by starting from the kinetic energy of a point particle (which is only translational), and building up a notion of the properties of systems of particles.

It turns out that the energy factors into a portion that has the $\frac{1}{2}Mv_\text{com}^2$ form (where $M$ is the total mass and $v_\text{com}$ is the velocity of the center of mass) and a portion due to motion of the parts relative the center of mass (which for a rigid body is the rotational kinetic energy).

This development is purely algebraic so you might be able to work it yourself with no more hint than I've given you, but it is also shown in every serious mechanics book.

dmckee --- ex-moderator kitten
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