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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

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# Mass of a Rotating Cylinder

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