For a uniform solid cylinder of radius R and length L , the radius of gyration about its central geometric axis is equal to its radius of gyration about an axis passing through its centre and perpendicular to its length. The ratio L R is:

Step-by-step solution

The moment of inertia of a uniform solid cylinder about its central geometric axis is I₁ = 1 2 MR^2 . Its radius of gyration about this axis is K₁ = I₁ M = R 2 . The moment of inertia of the cylinder about an axis passing through its centre and perpendicular to its length is I₂ = M ( R^2 4 + L^2 12 ) . Its radius of gyration about this axis is K₂ = I₂ M = R^2 4 + L^2 12 . Given that K₁ = K₂ , we have: K₁^2 = K₂^2 R^2 2 = R^2 4 + L^2 12 R^2 2 - R^2 4 = L^2 12 R^2 4 = L^2 12 L^2 = 3R^2 L R = 3 Answer: 3