CAIE M2 (Mechanics 2) 2016 November

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\includegraphics[max width=\textwidth, alt={}, center]{f9d76b90-9786-4a35-8f94-ffa7b18622b6-2_318_488_484_826} A uniform wire is bent to form an object which has a semicircular arc with diameter \(A B\) of length 1.2 m , with a smaller semicircular arc with diameter \(B C\) of length 0.6 m . The end \(C\) of the smaller arc is at the centre of the larger arc (see diagram). The two semicircular arcs of the wire are in the same plane.

1. Show that the distance of the centre of mass of the object from the line \(A C B\) is 0.191 m , correct to 3 significant figures. The object is freely suspended at \(A\) and hangs in equilibrium.
2. Find the angle between \(A C B\) and the vertical.

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\includegraphics[max width=\textwidth, alt={}, center]{f9d76b90-9786-4a35-8f94-ffa7b18622b6-3_388_650_264_749} The diagram shows the cross-section \(A B C D\) through the centre of mass of a uniform solid prism. \(A B = 0.9 \mathrm {~m} , B C = 2 a \mathrm {~m} , A D = a \mathrm {~m}\) and angle \(A B C =\) angle \(B A D = 90 ^ { \circ }\).

1. Calculate the distance of the centre of mass of the prism from \(A D\).
2. Express the distance of the centre of mass of the prism from \(A B\) in terms of \(a\). The prism has weight 18 N and rests in equilibrium on a rough horizontal surface, with \(A D\) in contact with the surface. A horizontal force of magnitude 6 N is applied to the prism. This force acts through the centre of mass in the direction \(B C\).
3. Given that the prism is on the point of toppling, calculate \(a\).

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\includegraphics[max width=\textwidth, alt={}, center]{f9d76b90-9786-4a35-8f94-ffa7b18622b6-3_483_419_1800_863} The diagram shows a smooth narrow tube formed into a fixed vertical circle with centre \(O\) and radius 0.9 m . A light elastic string with modulus of elasticity 8 N and natural length 1.2 m has one end attached to the highest point \(A\) on the inside of the tube. The other end of the string is attached to a particle \(P\) of mass 0.2 kg . The particle is released from rest at the lowest point on the inside of the tube. By considering energy, calculate

1. the speed of \(P\) when it is at the same horizontal level as \(O\),
2. the speed of \(P\) at the instant when the string becomes slack.