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Internal energy is all the energy of a system that is associated with its microscopic components (atoms and molecules) when viewed from a reference frame at rest with respect to the center of mass of the system.

That's the definition for the internal energy in the textbook I am studying from but I can't get exactly what's meant by the "reference frame at rest with respect to the center of mass of the system".

Is it the internal energy measured of the system measured from the equilibrium position (reference rest frame at rest).

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    $\begingroup$ The reason that they say "when viewed from a reference frame at rest with respect to the center of mass" is that any kinetic energy of the object as a whole is not considered to be part of its "internal energy". Viewing the system from a reference frame at rest with the center of mass effectively sets the kinetic energy of the object to zero. What you have left is the object's internal energy. $\endgroup$ – Samuel Weir Nov 5 '15 at 21:34
  • $\begingroup$ If we viewed the system at rest frame shouldn't the particles be at rest? $\endgroup$ – Abmon98 Nov 5 '15 at 21:52
  • $\begingroup$ No, there is always random thermal motion of all of the atoms and molecules. No frame of reference will make all of those motions zero. But you can always find a frame of reference in which the net motion of all of the atoms or molecules in any specific direction is zero. $\endgroup$ – Samuel Weir Nov 5 '15 at 21:55
  • $\begingroup$ The body itself is at rest state, but within that body are molecules which are in random motion. $\endgroup$ – Abmon98 Nov 5 '15 at 22:34
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The internal energy of a body is the summation of internal potential energy and internal kinetic energy.

Internal potential energy is the total potential energy because of the interactions between each pair of particles constituting the body.

Internal kinetic energy is the energy associated with the microscopic movements(rather motion) of the molecules of the body.

What your textbook tries to define is perhaps the internal kinetic energy. Because, roughly we can say that if in a frame of reference the center of mass of the body is not in motion then all the kinetic energy of the body in that frame will be due to the microscopic motion of its molecules - which is internal kinetic energy. But that is not actually a perfect definition for the internal kinetic energy. The reason is that if the body is not a perfectly rigid body then there may exist macroscopic motion of some parts of the body and still we can have the center of mass at rest. So all we can say about the internal kinetic energy is that it is the energy associated with the microscopic motions of the molecules of the body. Which is essentially a function of the temperature of the body.

The internal energy of a body remains the same independent of your frame of reference. (Neglecting relativistic effects, of course.) Because the internal potential energy depends upon the internal geometry of the body which is same in all the frames. The microscopic motion is more or less random and thus the kinetic energy content associated with this random motion is same in all the frames.

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  • $\begingroup$ Why do we refer to center of mass of the body, why cant we choose any point on your body as our reference $\endgroup$ – Abmon98 Nov 5 '15 at 22:35
  • $\begingroup$ Yes you can choose any point. Since the internal kinetic energy is due to random motion, it doesn't really matters which frame you are choosing. But again, if you are analyzing macroscopically rigid body and you choose a frame in which the rigid body is at rest then only the internal kinetic energy will be equal to total kinetic energy of the body in that frame. If you choose some other frame then internal kinetic energy still remains the same but the total kinetic energy changes because the macroscopic speeds of the body changes with frames. $\endgroup$ – Dvij Mankad Nov 6 '15 at 4:12

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