Moment of inertia of a cylinder of mass M, length L and radius R about an axis passing through its centre and perpendicular to the axis of the cylinder is  . If such a cylinder is to be made for a given mass of material, the ratio L/R for it to have minimum possible I is :-

The moment of inertia (I) of a cylinder about an axis passing through its center and perpendicular to the axis of the cylinder is given by the formula:

I = 1/2 * M * R^2

where M is the mass of the cylinder and R is the radius of the cylinder.

The volume (V) of the cylinder is given by the formula:

V = π * R^2 * L

where L is the length of the cylinder.

Given that the mass of the cylinder is constant, we can express the mass as:

M = ρ * V

where ρ is the density of the material of the cylinder.

Substituting the volume into the mass equation gives:

M = ρ * π * R^2 * L

Rearranging for L gives:

L = M / (ρ * π * R^2)

Substituting this into the moment of inertia equation gives:

I = 1/2 * M * R^2 = 1/2 * ρ * π * R^2 * L = 1/2 * ρ * π * R^2 * M / (ρ * π * R^2) = 1/2 * M^2 / (ρ * π * R^2)

To minimize the moment of inertia, we need to maximize the denominator, which is (ρ * π * R^2). Since ρ and π are constants, we need to maximize R^2, which means we need to maximize R.

Therefore, for the cylinder to have the minimum possible moment of inertia, the ratio L/R should be as small as possible. Since R should be as large as possible, L should be as small as possible. Therefore, the ratio L/R should approach 0 for the cylinder to have the minimum possible moment of inertia.