Question

A uniform hollow cylinder has a density p, a length 'L', an inner radius 'a', and an outer radius 'b', Its moment of inetia about the axis of the cylinder is (Mass of the cylinder is 'M')

• A. $M(b^2+a^2)$
• B. $2M(b^2+a^2)$
• C. $\frac{M}{2}(b^2+a^2)$
• D. $\frac{3M}{4}(b^2+a^2)$

Answer 1

The correct option is C $\frac{M}{2}(b^2+a^2)$

$\begin{array}{c}\text{A Hollow cylinder can be obtained by taking a solld}\\ \text{cylindrical cavity from a solid cylinder.}\end{array}$

$\begin{array}{cc}{I}_{\text{trollo}w}& =I_1-I_2\\ & =\frac{1}{2}\times m_1{b}^2-\frac{1}{2}{m}_2{a}^2\\ & =\frac{1}{2}\times P_x\pi b^2\times L\times b^2-\frac{1}{2}\times rho\times \pi a^2\times 1\times a^2\\ & =\frac{1}{2}P\pi ((b^2-a^2)(6^2+a^2)\end{array}$

$\begin{array}{c}\text{Given mass of hollow cylinder}=m\\ =p_x\pi (6^2-a^2)\times l\\ \Rightarrow I_{\text{hollow}}=\frac{M}{2}(b^2+a^2)\end{array}$