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I modified the problem of Irodov's Problems in General Physics 1.343 as below: A stationary compound solid consisting of a cone and a hemisphere on its circular base has a taper angle 45 degree and a lateral surface of 4 square metres. What's is the volume of the whole compound object relative to a frame which is moving with (4/5)c velocity along the axis of the cone.

I can easily do the volume for cone but for the hemisphere I don't know what to do.

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Since length contraction occurs only along direction of (relative) motion, while length along other directions remain unaltered, just multiply the volume of stationary object by the factor of contraction, $\gamma\equiv\frac{1}{\sqrt{1-v^2}}$, in the units where $c=1$. It doesn't matter what shape the object has.

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  • $\begingroup$ Ooops. I made a blunder. I wrote cylinder instead of cone. But I don't see how will your answer work here. $\endgroup$ – Mockingbird Sep 24 '16 at 7:15
  • $\begingroup$ I assumed you are able to find volume of the object in stationary frame, and wanted to know what it would be as seen from another moving inertial frame. $\endgroup$ – Deep Sep 24 '16 at 9:17
  • $\begingroup$ Well, can you give me proof of your answer? Besides your gamma factor gets a complex value. $\endgroup$ – Mockingbird Sep 24 '16 at 23:57
  • $\begingroup$ $c=1$, therefore $v=4/5$ for your problem. As far proof is concerned think of one axis, say X-axis contracting by $\gamma$ factor, so volume reduces by only that factor. $\endgroup$ – Deep Sep 25 '16 at 7:25
  • $\begingroup$ Is it true for area too? I mean if in the same situation except the cone be replaced by a isosceles triangle, will the area only contract by gamma factor? $\endgroup$ – Mockingbird Oct 14 '16 at 9:04

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