CAIE FP2 (Further Pure Mathematics 2) 2019 June

1 A particle \(P\) moves along an arc of a circle with centre \(O\) and radius 2 m . At time \(t\) seconds, the angle POA is \(\theta\), where \(\theta = 1 - \cos 2 t\), and \(A\) is a fixed point on the arc of the circle.

1. Show that the magnitude of the radial component of the acceleration of \(P\) when \(t = \frac { 1 } { 6 } \pi\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
   \includegraphics[max width=\textwidth, alt={}, center]{bbb7dbf7-1322-42d4-a0af-3850a4ea95ac-03_65_1573_488_324}
2. Find the magnitude of the transverse component of the acceleration of \(P\) when \(t = \frac { 1 } { 6 } \pi\).

2 A particle \(P\) moves on a straight line in simple harmonic motion. The centre of the motion is \(O\). The points \(A\) and \(B\) are on the line on opposite sides of \(O\) such that \(O A = 3.5 \mathrm {~m}\) and \(O B = 1 \mathrm {~m}\). The speed of \(P\) when it is at \(B\) is twice its speed when it is at \(A\). The maximum acceleration of \(P\) is \(1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the speed of \(P\) when it is at \(O\).
   \includegraphics[max width=\textwidth, alt={}, center]{bbb7dbf7-1322-42d4-a0af-3850a4ea95ac-04_64_1566_492_328}
2. Find the time taken by \(P\) to travel directly from \(A\) to \(B\).

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\includegraphics[max width=\textwidth, alt={}, center]{bbb7dbf7-1322-42d4-a0af-3850a4ea95ac-08_677_812_258_664} A uniform rod \(A B\) of length \(4 a\) and weight \(W\) rests with the end \(A\) in contact with a rough vertical wall. A light inextensible string of length \(\frac { 5 } { 2 } a\) has one end attached to the point \(C\) on the rod, where \(A C = \frac { 5 } { 2 } a\). The other end of the string is attached to a point \(D\) on the wall, vertically above \(A\). The vertical plane containing the \(\operatorname { rod } A B\) is perpendicular to the wall. The angle between the rod and the wall is \(\theta\), where \(\tan \theta = 2\) (see diagram). The end \(A\) of the rod is on the point of slipping down the wall and the coefficient of friction between the rod and the wall is \(\mu\). Find, in either order, the tension in the string and the value of \(\mu\).

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\includegraphics[max width=\textwidth, alt={}, center]{bbb7dbf7-1322-42d4-a0af-3850a4ea95ac-10_809_778_258_680} A thin uniform \(\operatorname { rod } A B\) has mass \(k M\) and length \(2 a\). The end \(A\) of the rod is rigidly attached to the surface of a uniform hollow sphere with centre \(O\), mass \(k M\) and radius \(2 a\). The end \(B\) of the rod is rigidly attached to the circumference of a uniform ring with centre \(C\), mass \(M\) and radius \(a\). The points \(C , B , A , O\) lie in a straight line. The horizontal axis \(L\) passes through the mid-point of the rod and is perpendicular to the rod and in the plane of the ring (see diagram). The object consisting of the rod, the ring and the hollow sphere can rotate freely about \(L\).

1. Show that the moment of inertia of the object about \(L\) is \(\frac { 3 } { 2 } ( 8 k + 3 ) M a ^ { 2 }\).
   The object performs small oscillations about \(L\), with the ring above the sphere as shown in the diagram.
2. Find the set of possible values of \(k\) and the period of these oscillations in terms of \(k\).