Mass of a Rotating Cylinder
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The mass
of a cylinder of radius R and height h, which is rotating at a constant
angular speed w,
can be found by integrating the infinitesimal mass element dm over the
volume of the cylinder. See Figure 1 below
It
should be noted that the proper mass density is not a function of speed in this
case. The mass element has the value
The
proper mass of the element is related to the proper mass density by
Thus Eq.
(1) becomes
The mass
of the rotating cylinder can now be found by integrating over the volume of the
cylinder. Using cylindrical coordinates
The
speed v of the mass element is related to the angular speed and the
radius r through v = wr.
Make the
substitution
Eq. (5)
now becomes
Noting
that M0
= r0V
= 2pRhr0
we obtain the final solution
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