Challenging +1.8 This is a challenging compound pendulum problem requiring: (1) calculation of moment of inertia for a composite body with non-uniform mass distribution (glass sheet plus four metal strips), using parallel axis theorem multiple times; (2) finding the center of mass of the composite system; (3) applying the small-angle approximation to derive SHM; (4) calculating the period. The multi-component setup, extensive algebraic manipulation, and integration of multiple mechanics concepts (rotational inertia, center of mass, SHM) make this substantially harder than typical A-level questions, though it follows a standard compound pendulum framework once set up.
5 A uniform rectangular thin sheet of glass \(A B C D\), in which \(A B = 8 a\) and \(B C = 6 a\), has mass \(\frac { 3 } { 5 } M\). Each of the edges \(A B , B C , C D\) and \(D A\) has a thin strip of metal attached to it, as a border to the glass. The strips along \(A B\) and \(C D\) each have mass \(M\), and the strips along \(B C\) and \(D A\) each have mass \(\frac { 1 } { 3 } M\). Show that the moment of inertia of the whole object (glass and metal strips) about an axis through \(A\) perpendicular to the plane of the object is \(128 M a ^ { 2 }\).
The object is free to rotate about this axis, which is fixed and smooth. The object hangs in equilibrium with \(C\) vertically below \(A\). It is displaced through a small angle and released from rest. Show that it will move in approximate simple harmonic motion and state the period of the motion.
5 A uniform rectangular thin sheet of glass $A B C D$, in which $A B = 8 a$ and $B C = 6 a$, has mass $\frac { 3 } { 5 } M$. Each of the edges $A B , B C , C D$ and $D A$ has a thin strip of metal attached to it, as a border to the glass. The strips along $A B$ and $C D$ each have mass $M$, and the strips along $B C$ and $D A$ each have mass $\frac { 1 } { 3 } M$. Show that the moment of inertia of the whole object (glass and metal strips) about an axis through $A$ perpendicular to the plane of the object is $128 M a ^ { 2 }$.
The object is free to rotate about this axis, which is fixed and smooth. The object hangs in equilibrium with $C$ vertically below $A$. It is displaced through a small angle and released from rest. Show that it will move in approximate simple harmonic motion and state the period of the motion.
\hfill \mbox{\textit{CAIE FP2 2014 Q5 [13]}}