Challenging +1.3 This is a 3D statics problem requiring knowledge of the center of mass of a cone (h/4 from base), setting up moment equilibrium about the suspension point, and solving a trigonometric equation. While it involves multiple steps and 3D geometry, the approach is standard for Further Maths mechanics: locate COM, take moments, solve. The given tan α = 1/3 simplifies calculations significantly. More challenging than typical A-level mechanics but routine for FM students who know cone COM formula.
\includegraphics{figure_2}
A uniform solid right circular cone has base radius \(a\) and semi-vertical angle \(\alpha\), where \(\tan \alpha = \frac{1}{3}\). The cone is freely suspended by a string attached at a point A on the rim of its base, and hangs in equilibrium with its axis of symmetry making an angle of \(\theta^0\) with the upward vertical, as shown in the diagram above.
Find, to one decimal place, the value of \(\theta\).
[5]
\includegraphics{figure_2}
A uniform solid right circular cone has base radius $a$ and semi-vertical angle $\alpha$, where $\tan \alpha = \frac{1}{3}$. The cone is freely suspended by a string attached at a point A on the rim of its base, and hangs in equilibrium with its axis of symmetry making an angle of $\theta^0$ with the upward vertical, as shown in the diagram above.
Find, to one decimal place, the value of $\theta$.
[5]
\hfill \mbox{\textit{SPS SPS FM Mechanics 2022 Q2 [5]}}