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Question
What can you infer if a uniform ring and a uniform disc have the same radius of gyration?
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Solution
The radius of gyration of a thin ring of radius Rr about its transverse symmetry axis is
`"k"_"r" = sqrt("I"_"CM"//"M"_"r") = sqrt("R"_"r"^2) = "R"_"r"`
The radius of gyration of a thin disc of radius `"R"_"d"` about its transverse symmetry axis is
`"k"_"d" = sqrt("I"_"CM"//"M"_"d") = (sqrt("M"_"d""R"_"d"^2)//2)/"M"_"d" = 1/sqrt2 "R"_"d"`
Given `"k"_"r" = "k"_"d"`
`"R"_"r" = 1/"R"_"d"` or, equivalently, `"R"_"d" = sqrt2 "R"_"r"`.
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