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Main Question or Discussion Point
##\ \ \ \ \ ##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 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 nonrotating 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’##.
##\ \ \ \ \ ##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 nonrotating 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’##.
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