Challenging +1.2 This is a standard centre of mass problem requiring students to find the centre of mass of a composite shape (triangle minus triangle), apply the equilibrium condition that the centre of mass must lie above the base, and solve an inequality. While it involves multiple steps (finding two centres of mass, using composite body formula, applying equilibrium condition), these are all routine techniques for Further Maths mechanics students with no novel insight required.
\includegraphics{figure_2}
A uniform lamina is in the form of a triangle \(ABC\) in which angle \(B\) is a right angle, \(AB = 9a\) and \(BC = 6a\). The point \(D\) is on \(BC\) such that \(BD = x\) (see diagram). The region \(ABD\) is removed from the lamina. The resulting shape \(ADC\) is placed with the edge \(DC\) on a horizontal surface and the plane \(ADC\) is vertical.
Find the set of values of \(x\), in terms of \(a\), for which the shape is in equilibrium.
[6]
\includegraphics{figure_2}
A uniform lamina is in the form of a triangle $ABC$ in which angle $B$ is a right angle, $AB = 9a$ and $BC = 6a$. The point $D$ is on $BC$ such that $BD = x$ (see diagram). The region $ABD$ is removed from the lamina. The resulting shape $ADC$ is placed with the edge $DC$ on a horizontal surface and the plane $ADC$ is vertical.
Find the set of values of $x$, in terms of $a$, for which the shape is in equilibrium.
[6]
\hfill \mbox{\textit{CAIE Further Paper 3 2022 Q2 [6]}}