Question

A cylinder of mass \(\mathrm{M}\) has length \(\mathrm{L}\) that is 3 times its radius what is the ratio of its moment of inertia about its own axis and that about an axis passing through its centre and perpendicular to its axis? \(\\{\mathrm{A}\\} 1\) \(\\{\mathrm{B}\\}(1 / \sqrt{3})\) \(\\{\mathrm{C}\\} \sqrt{3}\) \(\\{\mathrm{D}\\}(\sqrt{3} / 2)\)

Short answer

The short answer based on the step-by-step solution is: None of the given options match the correct ratio, which is \(\frac{3}{7}\).

Step-by-step solution

Moment of Inertia about its own axis

The moment of inertia, \(I_A\), of a cylinder about its own axis, assuming it is a solid cylinder, can be found using the following formula: \[ I_A = \frac{1}{2}MR^2, \] where \(M\) is the mass of the cylinder and \(R\) is its radius. In this case, \(R = r\). So, the moment of inertia about its own axis can be calculated as: \[ I_A = \frac{1}{2}Mr^2. \]

Moment of Inertia about the perpendicular axis

The moment of inertia of a cylinder about an axis passing through its center and perpendicular to its axis, \(I_B\), can be found using this formula: \[ I_B = \frac{1}{12}M(3d^2 + h^2), \] where d is the diameter of the cylinder which is \(2r\) (2 times the radius) and h is the height of the cylinder which is \(3r\), given the length is 3 times its radius. So, the moment of inertia about the perpendicular axis can be calculated as: \[ I_B = \frac{1}{12}M(3(2r)^2 + (3r)^2). \] Now, we have both moments of inertia \(I_A\) and \(I_B\). To find the ratio of these moments of inertia, we just need to divide them.

Finding the ratio

We will now find the ratio as follows: \[ \text{ratio} = \frac{I_A}{I_B} = \frac{\frac{1}{2}Mr^2}{\frac{1}{12}M(3(2r)^2 + (3r)^2)}. \] We can simplify the fraction, canceling out \(M\) and \(r^2\) from both the numerator and denominator and simplifying the expression: \[ \text{ratio}=\frac{\frac{1}{2}}{\frac{1}{12}(3(4) + 9)} = \frac{\frac{1}{2}}{\frac{1}{12}(12+9)} = \frac{\frac{1}{2}}{\frac{1}{12}(21)} = \frac{1}{2}\times\frac{12}{21}. \] Further simplifying the expression: \[ \text{ratio} = \frac{1}{2}\times\frac{6}{7} = \frac{6}{14} = \frac{3}{7}. \] Hence, none of the given options matches the correct ratio.