Hi assume that a Cylinder with radius [tex]\[ R \][/tex] , proper mass [tex]\[ M_0 \][/tex] and height [tex]\[ h \][/tex] which is rotating at a constant angular speed [tex]\[ \omega \][/tex] In order to calculate the relativistic mass we use the proper mass element to calculate the relativistic mass element , so : [tex]\[ dM = \frac{{dM_0 }}{{\sqrt {1 - \frac{{v^2 }}{{c^2 }}} }} \][/tex] But [tex]\[ dM_0 = \rho _0 dV_0 \][/tex] where [tex]\[ \rho _0 \][/tex] is the proper mass density and [tex]\[ dV_0 \][/tex] is the proper volume element . so : [tex]\[ \begin{array}{l} dM = \frac{{\rho _0 dV}}{{\sqrt {1 - \frac{{v^2 }}{{c^2 }}} }} \\ M = \int\limits_V {\frac{{\rho _0 dV}}{{\sqrt {1 - \frac{{v^2 }}{{c^2 }}} }}} = \int\limits_0^R {\int\limits_0^{2\pi } {\int\limits_0^h {\frac{{\rho _0 rdrd\phi dz}}{{\sqrt {1 - \frac{{v^2 }}{{c^2 }}} }}} } } = 2\pi h\rho _0 \int\limits_0^R {\frac{{rdr}}{{\sqrt {1 - \frac{{v^2 }}{{c^2 }}} }}} \\ but:v = \omega r \\ M = 2\pi h\rho _0 \int\limits_0^R {\frac{{rdr}}{{\sqrt {1 - \frac{{\omega ^2 }}{{c^2 }}r^2 } }}} \\ \end{array} \][/tex] Now , make the substitution [tex]\[ u = 1 - \frac{{\omega ^2 }}{{c^2 }}r^2 \Rightarrow du = - 2\frac{{\omega ^2 }}{{c^2 }}rdr \Rightarrow 2rdr = - \frac{{c^2 }}{{\omega ^2 }}du \][/tex] so : [tex]\[ \begin{array}{l} M = - \frac{{\pi h\rho _0 c^2 }}{{\omega ^2 }}\int\limits_1^{1 - \left( {\frac{{\omega R}}{c}} \right)^2 } {\frac{{du}}{{\sqrt u }}} = - \frac{{2\pi h\rho _0 c^2 }}{{\omega ^2 }}\left[ {\sqrt u } \right]_1^{1 - \left( {\frac{{\omega R}}{c}} \right)^2 } = \frac{{2\pi h\rho _0 c^2 }}{{\omega ^2 }}\left( {1 - \sqrt {1 - \left( {\frac{{\omega R}}{c}} \right)^2 } } \right) \\ but:M_0 = \rho _0 V_0 = \pi R^2 h\rho _0 \\ M = \frac{{2M_0 c^2 }}{{R^2 \omega ^2 }}\left( {1 - \sqrt {1 - \left( {\frac{{\omega R}}{c}} \right)^2 } } \right) \\ \end{array} \][/tex] now , there is something make me confused in this equation . If we put [tex]\[ \omega R = c \][/tex] we find that the relativistic mass is [tex]\[ M = 2M_0 \][/tex] . How it can be ?????? I know that any thing has a v = c it's mass goes to infinity . Again , How it can be ???????? Thanks
Suggest a hint Hi, one, welcome to PF! Here's a hint: can you think of kinematical effects other than the increase of the kinetic energy of elements of matter in your cylindrical slug (depending upon radial coordinate, for fixed omega) which might be relevant here? Some points to note: your computation occurs in flat spacetime, so this is a question about "rigid rotation" for constant angular velocity in str, not gtr, your (flat spacetime) computation assumes "constant density" on each circle of constant radius r where [itex]0 < r < r_0, \, 0 < z < h[/itex], but you can think about what happens in the frame of rim riding observers at [itex]r=r_0[/itex], suggest rewriting in terms of energy-momentum tensor and computing both alleged mass+kinetic energy (taking account of rotational KE of matter elements in the slug) and angular momentum, and taking small omega limit to see if results appear to have expected slow rotation limit (what are mass, rotational kinetical energy, angular momentum according to Newton in this situation?), due to difficulties in modeling spin-up (which must be nonrigid), it is not neccessarily straightforward to compare objects which are allegedly "equivalent except for angular velocity", if you demand "simple answers to simple questions", you are plumb outta luck here. These points and many more are discussed in previous rotating disk and rotating ring threads at PF.
*Some* of the previous threads are: https://www.physicsforums.com/showthread.php?t=181352 https://www.physicsforums.com/showthread.php?p=1520416 Basically, you've ignored some important relativistic effects. I have to go right now, I might be able to amplify on this a bit later.
It's probably mentioned in the previous threads, but The rigid rotating disk in relativity is one of the sci.physics.faqs on this famous problem. Rigid bodies are generally suspect in SR and GR, but in the particular case of the rotating disk, rigid bodies are mathematically impossible. This surprising result is known as Ehrenfest's paradox. It's not a true "paradox", it's just gotten that name because the result is surprising. Spinning disks are certainly possible, but they must be non-rigid, and this requires a much more sophisticated analysis to handle properly. Work is done when the disk deforms, and the amount of work can contribute appreciably to the energy (which some posters may regard as "relativistic mass") of the disk in the idealized case. In case of disks made of actual matter, they will break first before the relativistic effects become important. Some of the above threads contain links to attempts (Egan's and mine) to analyze non-rigid rotating disks and/or hoops. The analysis is highly complex and technical, and isn't peer-reviewed for possible errors. Unfortunately there doesn't appear to be any similar peer-reviewed examples in the literature, though one can find some peer-reviewed papers on rotating stars. Unfortunately these papers on rotating stars are rather advanced full GR treatments, not pedagogical SR treatments.
Hi Chris Hillman I tried to think in different ways but I didn't arrive at any thing. So can you tell me how to fix this problem thanks