Question

B-3 solve

QB-3. Find the moment of inertia of a uniform half-disc about an axis perpendicular to the plane and passing through its centre of mass. Mass of this disc is M and radius is R.

Answer 1

Dear Student,
$$\begin{gathered} HeremassofthehalfdiskisM \\ thenmassperunitarea\left(\sigma \right)=\frac{2M}{\pi R^2} \\ Soletastripatradiusrofdrthickness \\ soareaofthisstrip=\pi rdr \\ now \\ momentofinertiaofthisstrip=dI=\pi rdr\sigma r^2 \\ sototalinertiaofthehalfdisk={\int }_0^R\pi rdr\sigma r^2={\int }_0^R\frac{2M}{R^2}{r}^{3}dr=\frac{M{R}^2}{2} \end{gathered}$$
Regard