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Relativistic Mass is: $$ m_r = \frac{m}{\sqrt{1 - v^2/c^2}} $$

So Einstein says that the faster an object moves, the more mass it gains (relativistic mass). So suppose you have a spherical ball with a radius of 10 meters and a resting mass of 100kg that is spinning on an axis. In fact, this ball is spinning so fast that any point on the equator is traveling at $0.9c$ around the axis. What is the relativistic mass of the ball when it is spinning at this speed?

I thunk up an integral that might describe it but I don't know. (based on a cartesian plane.) The equation for the velocity at any given point is (Revolutions/s*2piRadius)=Velocity. Radius = sqroot(x^2+y^2+z^2). RadiusRevolutions/s*2pi=v. Triple Integral replacing that equation for v of the relative mass equation?

Kaden Hunter
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  • Lol you think I do this for homework I'm in highschool man. No but really I don't even know where to start with to how to solve this problem. Because you have to find the function for how the relatistic mass changes as a function of R(radius) but then you have to take in account that m0 value changes even on the same radius because all the points are moving different speeds. Man just be cool I wanna see someone solve this and how they did it I have a huge interest in math and physics and this was a question that I thought of so just be cool. – Kaden Hunter Jan 21 '15 at 06:31
  • Here's the hint on how to start: you have to split up the mass into tiny regions with mass dm, and then integrate over them, so you'll have $M = \int \frac{dm}{\sqrt{1-v^{2}}}$. If you do this in cylindrical coordinates, $dm = (M/V)2\pi r dr dz$ and $v(r) = \omega r$, and the rest is just calculus. – Zo the Relativist Jan 21 '15 at 06:42
  • Now, the reason why people aren't given this as homework is the calculus is a bit non-trivial. – Zo the Relativist Jan 21 '15 at 06:46
  • Right, I don't expect the answer to be expressible in terms of simple transcendental functions (though it may be) – Zo the Relativist Jan 21 '15 at 06:55
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    I don't think this should have been closed. It strikes me as an exceedingly difficult question. I'm not sure the relativistic motion of rigid bodies is fully understood even now (it certainly isn't fully understood by me). The full solution would have to include not just the energy of rotation but also energy due to internal stresses. If an answer exists I would be very interested to see it. – John Rennie Jan 21 '15 at 07:05
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    @JohnRennie difficult question or not, I strongly believe it is off topic, since Kaden Hunter is just asking us to solve a problem, not asking a conceptual question. This site is not a problem-solving service. Neither is it a puzzle community where members challenge each other to solve problems. I can certainly agree that the problem may be difficult, and that one could ask a good question about how to solve it, but just challenging people to provide a full solution is entirely the wrong way to go about it. – David Z Jan 21 '15 at 08:57
  • Certainly related: the Ehrenfest Paradox. – Kieran Hunt Jan 21 '15 at 09:02
  • I can see quite a bit of conceptual stuff in the second to last paragraph. – N. Virgo Jan 21 '15 at 10:05
  • Well,this may help you:According to your assumption,relativistic mass is 127.629kg(for more,at 0.99c:144.904kg,at 0.999c:149.158kg,at c:150kg).PS:I did it with Mathematica. – Noctihaar Surralyn Jan 21 '15 at 10:13
  • Related: http://physics.stackexchange.com/q/8659/2451 and links therein. – Qmechanic Nov 05 '15 at 11:33
  • Suggestion to the title (v13): Instead of asking for the relativistic mass (which is an obsolete notion), ask about the total energy of the disc. – Qmechanic Nov 05 '15 at 14:27

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