2.8 Combined Translational and Spin Motions - Relativistic Mass.When a spinning spherical mass also possesses significant translational motion, the situation becomes a little more complex. Only the mass of the body will be analysed under this condition. Other parameters will exhibit similar characteristics. Consider a plan view of Fig. 2.1  From Fig. 2.4 the spin velocity of the elemental mass, normal to the direction of the translational motion is
where
The spin velocity in the same direction as the translational velocity is
The total translational velocity of the elemental is then the relativistic sum of (2.44) and v, i.e.
The magnitude of the total velocity of the elemental is then
The mass of the elemental at rest is given by (2.2) so that the energy mass of the elemental in this scenario is
First, take a relativistic approximation of the second term of the denominator retaining only second order terms. This gives simply
Now take a second relativistic approximation again retaining only second order terms. Thus
Integrating (2.49) with respect to f gives the energy mass of the toroid
Integrating (2.51) with respect to j gives the energy mass of the spherical shell, thus
Finally integrating (2.53) with respect to r gives the energy mass of the sphere thus
Converting this to a standard relativistic mass equation via an approximate binomial contraction gives
For the purpose of determining the energy mass of a spinning spherical body also possessing a linear motion, it therefore appears that, in a second order relativistic approximation, the two separate velocities, each at their own individual mass effective dimension, are simply added together as orthogonal vectors. Note that because of this combination of the two velocities, the limits to which each can be theoretically increased is reduced. In particular, if v is large enough, it would not be possible to increase w such that wG® c.
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P.G.Bass, June 2006
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