# Shown in the figure is a hollow ice cream cone (it is open at the top). If its mass is 'M', radius of its top, R and height H, then its M.O.I about its axis is : Option: 1 Option: 2 Option: 3 Option: 4

We will take an elemental ring of thickness dy and radius r at a slant height y as show in the figure

$\begin{array}{l}\\ given, \ \operatorname{Mass}=M \\ \operatorname{Radius}=R \\ I_{H C}=\int d I \operatorname{ring}=\int(d m) r^{2} \\ \\ d m=\frac{M}{\pi R \l}(2 \pi r d y) \end{array}$

where dm is the mass of the elemental ring

$I_{H C}= \int \frac{2 M}{R \ell} r^{3} d y$

$\frac{r}{R}=\frac{y}{l} \quad \Rightarrow \quad r=\frac{R}{l}y$

$I_{H C}=\frac{2 M}{Rl} \frac{R^{3}}{l^{3}} \int_{0}^{l} y^{3} d y=\frac{MR^2}{2}$

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