Challenging +1.8 This is a challenging Further Maths mechanics problem requiring multiple sophisticated techniques: calculating moment of inertia using parallel axis theorem for a composite lamina, finding the center of mass of an irregular shape, and applying energy conservation for rotational motion with a non-trivial inequality condition. While the steps are systematic, the problem demands careful geometric reasoning, algebraic manipulation, and understanding of when complete revolutions occur (minimum speed at top position). This is significantly harder than standard A-level mechanics but follows established FM2 patterns.
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\(A B C D\) is a uniform rectangular lamina of mass \(m\) in which \(A B = 4 a\) and \(B C = 2 a\). The lines \(A C\) and \(B D\) intersect at \(O\). The mid-points of \(O A , O B , O C , O D\) are \(E , F , G , H\) respectively. The rectangle \(E F G H\), in which \(E F = 2 a\) and \(F G = a\), is removed from \(A B C D\) (see diagram). The resulting lamina is free to rotate in a vertical plane about a fixed smooth horizontal axis through \(A\) and perpendicular to the plane of \(A B C D\). Show that the moment of inertia of this lamina about the axis is \(\frac { 85 } { 16 } m a ^ { 2 }\).
The lamina hangs in equilibrium under gravity with \(C\) vertically below \(A\). The point \(C\) is now given a speed \(u\). Given that the lamina performs complete revolutions, show that
$$u ^ { 2 } > \frac { 192 \sqrt { } 5 } { 17 } a g .$$
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$A B C D$ is a uniform rectangular lamina of mass $m$ in which $A B = 4 a$ and $B C = 2 a$. The lines $A C$ and $B D$ intersect at $O$. The mid-points of $O A , O B , O C , O D$ are $E , F , G , H$ respectively. The rectangle $E F G H$, in which $E F = 2 a$ and $F G = a$, is removed from $A B C D$ (see diagram). The resulting lamina is free to rotate in a vertical plane about a fixed smooth horizontal axis through $A$ and perpendicular to the plane of $A B C D$. Show that the moment of inertia of this lamina about the axis is $\frac { 85 } { 16 } m a ^ { 2 }$.
The lamina hangs in equilibrium under gravity with $C$ vertically below $A$. The point $C$ is now given a speed $u$. Given that the lamina performs complete revolutions, show that
$$u ^ { 2 } > \frac { 192 \sqrt { } 5 } { 17 } a g .$$
\hfill \mbox{\textit{CAIE FP2 2013 Q5 [13]}}