A uniform solid cylinder of mass M , length L and radius R has the same radius of gyration about its central longitudinal axis as it does about an axis passing through its centre and perpendicular to its length. The ratio of its length to its radius ( L R ) is:

Step-by-step solution

The moment of inertia of a uniform solid cylinder of mass M , radius R and length L about its central longitudinal axis is: I₁ = MR^2 2 Its radius of gyration about this axis is k₁ = I₁ M = R 2 . The moment of inertia of the same 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 = R^2 4 + L^2 12 . Given that k₁ = k₂ , we can equate their squares: R^2 2 = R^2 4 + L^2 12 R^2 2 - R^2 4 = L^2 12 R^2 4 = L^2 12 3R^