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##\ \ \ \ \ ##Calculate the 4 momentum of a rotating rod. We divide it into 4 parts. The part 1 is the work of predecessors.

##\ \ \ \ \ ##In Special relativity, the motion of rod

##\ \ \ \ \ ##In inertial reference frame ##K##, an isolated rod AB rotates around its mid point O at uniform angular velocityω, keeps the linear state.

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

##\ \ \ \ \ ##The inertial reference frame ##K’## moves relative to ##K##at speed##v##. Relative to##K’##, the auxiliary circle D and M appear as ellipses D and M, and they move at speed##-v##.

##\ \ \ \ \ ##Relative to##K’##, rod AB can't keep the linear state at all times. The shape of rotating wheel relative to ##K’## has been studied by the follow reference.

[1] A relativistic trolley paradox, Vadim N. Matvejev, Oleg V. Matvejev, and Ø. Grøn, American Journal of Physics 84, 419 (2016).

[2] A Rotating Disk in Translation, http://www.mathpages.com/home/kmath197/kmath197.htm.

[3] Appearance of a Relativistically Rotating Disk, Keith MeFarlane, International Journal of Theoretical Physics, Vol.20. No.6, 1981.

Fig.8. The non-rotating disk is drawn to the left and the rolling disk to the right. Radial lines of the rolling disk are curved as observed by simultaneity in the frame##K’##

##\ \ \ \ \ ##They have considered a wheel or a disk. Now we consider the momentum of a spokes on the wheel.

##\ \ \ \ \ ##With parameter##r_A=100m##, ##ω=-2.7E6rad/s##,##v=-0.9c##, using numerical methods, get the shape of

##\ \ \ \ \ ##In Special relativity, the motion of rod

*AB*(which is an object in non inertial motion) can be described in an inertial reference frame and the motion of rod*AB*can be converted to be described in another inertial reference system, by Lorenz transformation.##\ \ \ \ \ ##In inertial reference frame ##K##, an isolated rod AB rotates around its mid point O at uniform angular velocityω, keeps the linear state.

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

##\ \ \ \ \ ##The inertial reference frame ##K’## moves relative to ##K##at speed##v##. Relative to##K’##, the auxiliary circle D and M appear as ellipses D and M, and they move at speed##-v##.

##\ \ \ \ \ ##Relative to##K’##, rod AB can't keep the linear state at all times. The shape of rotating wheel relative to ##K’## has been studied by the follow reference.

[1] A relativistic trolley paradox, Vadim N. Matvejev, Oleg V. Matvejev, and Ø. Grøn, American Journal of Physics 84, 419 (2016).

[2] A Rotating Disk in Translation, http://www.mathpages.com/home/kmath197/kmath197.htm.

[3] Appearance of a Relativistically Rotating Disk, Keith MeFarlane, International Journal of Theoretical Physics, Vol.20. No.6, 1981.

Fig.8. The non-rotating disk is drawn to the left and the rolling disk to the right. Radial lines of the rolling disk are curved as observed by simultaneity in the frame##K’##

##\ \ \ \ \ ##They have considered a wheel or a disk. Now we consider the momentum of a spokes on the wheel.

##\ \ \ \ \ ##With parameter##r_A=100m##, ##ω=-2.7E6rad/s##,##v=-0.9c##, using numerical methods, get the shape of

*AB*observed in ##K’##.