Challenging +1.8 This question requires applying parallel axis theorem to an annular lamina (non-trivial geometry), setting up energy conservation with rotational KE, and solving an inequality for the angular speed range. It combines multiple advanced mechanics concepts but follows a structured path with clear physical setup, making it challenging but accessible to well-prepared Further Maths students.
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A uniform lamina of mass \(m\) is bounded by concentric circles with centre \(O\) and radii \(a\) and \(2 a\). The lamina is free to rotate about a fixed smooth horizontal axis \(T\) which is tangential to the outer rim (see diagram). Show that the moment of inertia of the lamina about \(T\) is \(\frac { 21 } { 4 } m a ^ { 2 }\).
When hanging at rest, with \(O\) vertically below \(T\), the lamina is given an angular speed \(\omega\) about \(T\). The lamina comes to instantaneous rest in the subsequent motion. Neglecting air resistance, find the set of possible values of \(\omega\).
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A uniform lamina of mass $m$ is bounded by concentric circles with centre $O$ and radii $a$ and $2 a$. The lamina is free to rotate about a fixed smooth horizontal axis $T$ which is tangential to the outer rim (see diagram). Show that the moment of inertia of the lamina about $T$ is $\frac { 21 } { 4 } m a ^ { 2 }$.
When hanging at rest, with $O$ vertically below $T$, the lamina is given an angular speed $\omega$ about $T$. The lamina comes to instantaneous rest in the subsequent motion. Neglecting air resistance, find the set of possible values of $\omega$.
\hfill \mbox{\textit{CAIE FP2 2009 Q3 [8]}}