A ring and a solid sphere, rotating about axes passing through their centres, have the same radius of gyration. The axis of rotation of the ring is perpendicular to its plane. The ratio of the radius of the ring to that of the sphere is √2/x. The value of x is .

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5. Radius of gyration is defined by I=Mk², so k²=I/M. Ring about the perpendicular axis through its centre: I=MRᵣ², so k²=Rᵣ². Solid sphere about a diameter: I=2/5MRₛ², so k²=2/5Rₛ². Setting the two equal: Rᵣ²=2/5Rₛ². (Rᵣ)/(Rₛ)=√2/5, so x=5. The masses never appear — radius of gyration is a statement about shape alone.

A ring and a solid sphere, rotating about axes passing through their centres, have the same radius of gyration. The axis of rotation of the ring is perpendicular to its plane. The ratio of the radius of the ring to that of the sphere is √2/x. The value of x is ______.

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