Challenging +1.2 This is a standard M5 moment of inertia derivation using the disc formula provided. Students must set up the integral with variable radius r(x), express dm in terms of dx using density, and integrate. While it requires careful algebra and multiple steps (10 marks), the method is well-practiced and follows a template taught explicitly in M5 courses. It's harder than routine mechanics but not exceptionally challenging for this advanced module.
Prove, using integration, that the moment of inertia of a uniform solid right circular cone, of mass \(M\) and base radius \(a\), about its axis is \(\frac{3}{10}Ma^2\).
[You may assume, without proof, that the moment of inertia of a uniform circular disc, of mass \(m\) and radius \(r\), about an axis through its centre and perpendicular to its plane is \(\frac{1}{2}mr^2\).]
[10]
Prove, using integration, that the moment of inertia of a uniform solid right circular cone, of mass $M$ and base radius $a$, about its axis is $\frac{3}{10}Ma^2$.
[You may assume, without proof, that the moment of inertia of a uniform circular disc, of mass $m$ and radius $r$, about an axis through its centre and perpendicular to its plane is $\frac{1}{2}mr^2$.]
[10]
\hfill \mbox{\textit{Edexcel M5 Q7 [10]}}