Challenging +1.2 This is a standard centre of mass problem for a composite body requiring the given formula, volume calculations for cones, and the composite body formula. While it involves multiple steps and algebraic manipulation to reach the specific result, the method is entirely routine for Further Maths students who have practiced similar frustum problems. The 'show that' format provides a target to work towards, reducing problem-solving demand.
2. You are given that the centre of mass of a uniform solid cone of height \(h\) and base radius \(r\) is at a height of \(\frac { 1 } { 4 } h\) above its base.
The diagram shows a solid conical frustum. It is formed by taking a uniform right circular cone, of base radius \(3 x\) and height \(6 y\), and removing a smaller cone, of base radius \(x\), with the same vertex.
\includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-3_490_903_1937_575}
Show that the distance of the centre of mass of the frustum from its base along the axis of symmetry is \(\frac { 18 } { 13 } y\).
2. You are given that the centre of mass of a uniform solid cone of height $h$ and base radius $r$ is at a height of $\frac { 1 } { 4 } h$ above its base.
The diagram shows a solid conical frustum. It is formed by taking a uniform right circular cone, of base radius $3 x$ and height $6 y$, and removing a smaller cone, of base radius $x$, with the same vertex.\\
\includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-3_490_903_1937_575}
Show that the distance of the centre of mass of the frustum from its base along the axis of symmetry is $\frac { 18 } { 13 } y$.\\
\hfill \mbox{\textit{WJEC Further Unit 6 2023 Q2 [7]}}