Standard +0.8 This is a standard Further Maths mechanics question requiring (a) a derivation by integration of a known result for hemisphere centre of mass, and (b) application to a composite body with different densities. While it involves multiple steps and careful bookkeeping with the density ratio, it follows a well-established template for composite centre of mass problems without requiring novel insight.
5. (a) Show, by integration, that the centre of mass of a uniform solid hemisphere of radius \(r\) is at a distance of \(\frac { 3 r } { 8 }\) from the plane face.
(b) The diagram shows a composite solid body which consists of a uniform right circular cylinder capped by a uniform hemisphere. The total height of the solid is \(3 r \mathrm {~cm}\), where \(r\) represents the common radius of the hemisphere and the cylinder.
\includegraphics[max width=\textwidth, alt={}, center]{3578a810-46da-4d9e-a98f-248be72a517a-6_397_340_762_858}
Given that the density of the hemisphere is \(50 \%\) more than that of the cylinder, find the distance of the centre of mass of the solid from its base along the axis of symmetry.
5. (a) Show, by integration, that the centre of mass of a uniform solid hemisphere of radius $r$ is at a distance of $\frac { 3 r } { 8 }$ from the plane face.\\
(b) The diagram shows a composite solid body which consists of a uniform right circular cylinder capped by a uniform hemisphere. The total height of the solid is $3 r \mathrm {~cm}$, where $r$ represents the common radius of the hemisphere and the cylinder.\\
\includegraphics[max width=\textwidth, alt={}, center]{3578a810-46da-4d9e-a98f-248be72a517a-6_397_340_762_858}
Given that the density of the hemisphere is $50 \%$ more than that of the cylinder, find the distance of the centre of mass of the solid from its base along the axis of symmetry.\\
\hfill \mbox{\textit{WJEC Further Unit 6 2019 Q5 [10]}}