\(\displaystyle \ \ \ \ \ \)The sum of rest masses of a born rigid body consists of two parts: the potential energy of a born rigid body, and the actual particles or the material in the form of physical existence.

\(\displaystyle \ \ \ \ \ \) For a born rigid body, there is no conversion between matter and energy. The actual particles or the material in the form of physical existence of a rigid body is constant. So , if the potential energy of born rigid body is a constant, the sum of rest masses of a born rigid body is a constant.

\(\displaystyle \ \ \ \ \ \)If rod \(\displaystyle AB \) is a Born rigid body, sum of rest masses of every section \(\displaystyle DM \) of rod \(\displaystyle AB \) is a constant measured in any inertial reference frame. There are \(\displaystyle { m_{sum\ 0 }({t_1} ’, DM) } ’ = { m_{sum\ 0 }({t_2} ’, DM) } ’ \) (in \(\displaystyle K’\)),=\(\displaystyle { m_{sum\ 0 }({t_3} , DM) } \) （ in \(\displaystyle K\) ）.

\(\displaystyle \ \ \ \ \ \)So I want to know whether “The potential energy of born rigid body is a constant measured in an inertial reference frame at different time?”, the born rigid body may be in non-inertial motion.

\(\displaystyle \ \ \ \ \ \) We consider a born rigid body model.

\(\displaystyle \ \ \ \ \ \) In \(\displaystyle K\) , an isolated rod \(\displaystyle AB \) rotates around its mid point \(\displaystyle O \) at uniform angular velocity \(\displaystyle ω\) , keeps the linear state. For actual rod \(\displaystyle AB \), there is micro motion on molecular level. In the process of simplifying and establishing a mathematical model, we do not consider this micro motion. The rod \(\displaystyle AB \) is an ideal system without any relative movement between the components relative to \(\displaystyle K\) . The distribution of material and potential energy on the rod \(\displaystyle AB \) along the length is symmetry \(\displaystyle AB \) out mid point \(\displaystyle O \) and it does not change with time, measured in \(\displaystyle K\) . Rod \(\displaystyle AB \) has no thickness along \(\displaystyle θ \) direction, if transform \(\displaystyle K\) from rectangular coordinate system to polar coordinate system. That is to say consider rod \(\displaystyle AB \) as a one-dimensional rod on \(\displaystyle x-y\) plane relative to \(\displaystyle K\) .

\(\displaystyle \ \ \ \ \ \) For each point on \(\displaystyle AB \), the distance from point \(\displaystyle O \) to this point measured in \(\displaystyle K\) is constant; denote it is \(\displaystyle r\) . Such as for point \(\displaystyle D \)，\(\displaystyle r_D\) is a constant. For each point on \(\displaystyle OA \), its equation of motion is \(\displaystyle x=r\cos(ωt)\) , \(\displaystyle y=r\sin(ωt)\) . For each point on \(\displaystyle OB \), its equation of motion is \(\displaystyle x=r\cos(ωt+π)\) , \(\displaystyle y=r\sin(ωt+π)\) .

\(\displaystyle \ \ \ \ \ \)The inertial reference frame \(\displaystyle K’\) moves relative to \(\displaystyle K\) at speed \(\displaystyle v\).

\(\displaystyle \ \ \ \ \ \) For a born rigid body, there is no conversion between matter and energy. The actual particles or the material in the form of physical existence of a rigid body is constant. So , if the potential energy of born rigid body is a constant, the sum of rest masses of a born rigid body is a constant.

\(\displaystyle \ \ \ \ \ \)If rod \(\displaystyle AB \) is a Born rigid body, sum of rest masses of every section \(\displaystyle DM \) of rod \(\displaystyle AB \) is a constant measured in any inertial reference frame. There are \(\displaystyle { m_{sum\ 0 }({t_1} ’, DM) } ’ = { m_{sum\ 0 }({t_2} ’, DM) } ’ \) (in \(\displaystyle K’\)),=\(\displaystyle { m_{sum\ 0 }({t_3} , DM) } \) （ in \(\displaystyle K\) ）.

\(\displaystyle \ \ \ \ \ \)So I want to know whether “The potential energy of born rigid body is a constant measured in an inertial reference frame at different time?”, the born rigid body may be in non-inertial motion.

\(\displaystyle \ \ \ \ \ \) We consider a born rigid body model.

\(\displaystyle \ \ \ \ \ \) In \(\displaystyle K\) , an isolated rod \(\displaystyle AB \) rotates around its mid point \(\displaystyle O \) at uniform angular velocity \(\displaystyle ω\) , keeps the linear state. For actual rod \(\displaystyle AB \), there is micro motion on molecular level. In the process of simplifying and establishing a mathematical model, we do not consider this micro motion. The rod \(\displaystyle AB \) is an ideal system without any relative movement between the components relative to \(\displaystyle K\) . The distribution of material and potential energy on the rod \(\displaystyle AB \) along the length is symmetry \(\displaystyle AB \) out mid point \(\displaystyle O \) and it does not change with time, measured in \(\displaystyle K\) . Rod \(\displaystyle AB \) has no thickness along \(\displaystyle θ \) direction, if transform \(\displaystyle K\) from rectangular coordinate system to polar coordinate system. That is to say consider rod \(\displaystyle AB \) as a one-dimensional rod on \(\displaystyle x-y\) plane relative to \(\displaystyle K\) .

\(\displaystyle \ \ \ \ \ \) For each point on \(\displaystyle AB \), the distance from point \(\displaystyle O \) to this point measured in \(\displaystyle K\) is constant; denote it is \(\displaystyle r\) . Such as for point \(\displaystyle D \)，\(\displaystyle r_D\) is a constant. For each point on \(\displaystyle OA \), its equation of motion is \(\displaystyle x=r\cos(ωt)\) , \(\displaystyle y=r\sin(ωt)\) . For each point on \(\displaystyle OB \), its equation of motion is \(\displaystyle x=r\cos(ωt+π)\) , \(\displaystyle y=r\sin(ωt+π)\) .

\(\displaystyle \ \ \ \ \ \)The inertial reference frame \(\displaystyle K’\) moves relative to \(\displaystyle K\) at speed \(\displaystyle v\).

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