CAIE FP2 (Further Pure Mathematics 2) 2016 November

1
\includegraphics[max width=\textwidth, alt={}, center]{184020e1-7ff2-4172-8d33-baff963afa76-2_125_641_262_751} The point \(C\) is on the fixed line \(l\). Points \(A\) and \(B\) on \(l\) are such that \(A C = 4 \mathrm {~m}\) and \(C B = 2 \mathrm {~m}\), with \(C\) between \(A\) and \(B\). The point \(M\) is the mid-point of \(A B\) (see diagram). A particle \(P\) of mass \(m\) oscillates between \(A\) and \(B\) in simple harmonic motion. When \(P\) is at \(C\), its speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the magnitude of the maximum acceleration of \(P\),
2. the number of complete oscillations made by \(P\) in one minute,
3. the time that \(P\) takes to travel directly from \(A\) to \(C\).

2
\includegraphics[max width=\textwidth, alt={}, center]{184020e1-7ff2-4172-8d33-baff963afa76-2_531_760_927_696} Two smooth vertical walls each with their base on a smooth horizontal surface intersect at an angle of \(60 ^ { \circ }\). A small smooth sphere \(P\) is moving on the horizontal surface with speed \(u\) when it collides with the first vertical wall at the point \(D\). The angle between the direction of motion of \(P\) and the wall is \(\alpha ^ { \circ }\) before the collision and \(75 ^ { \circ }\) after the collision. The speed of \(P\) after this collision is \(v\) and the coefficient of restitution between \(P\) and the first wall is \(e\). Sphere \(P\) then collides with the second vertical wall at the point \(E\). The speed of \(P\) after this second collision is \(\frac { 1 } { 4 } u\) (see diagram). The coefficient of restitution between \(P\) and the second wall is \(\frac { 3 } { 4 }\).

1. By considering the collision at \(E\), show that \(v = \frac { \sqrt { } 2 } { 5 } u\).
2. Find the value of \(\alpha\) and the value of \(e\).

3
\includegraphics[max width=\textwidth, alt={}, center]{184020e1-7ff2-4172-8d33-baff963afa76-3_898_1116_258_518} The end \(P\) of a uniform rod \(P Q\), of weight \(k W\) and length \(8 a\), is rigidly attached to a point on the surface of a uniform sphere with centre \(C\), weight \(W\) and radius \(a\). The end \(Q\) is rigidly attached to a point on the surface of an identical sphere with centre \(D\). The points \(C , P , Q\) and \(D\) are in a straight line. The object consisting of the rod and two spheres rests with one sphere in contact with a rough horizontal surface, at the point \(A\), and the other sphere in contact with a smooth vertical wall, at the point \(B\). The angle between \(C D\) and the horizontal is \(\theta\). The point \(B\) is at a height of \(7 a\) above the base of the wall (see diagram). The points \(A , B , C , D , P\) and \(Q\) are all in the same vertical plane.

1. Show that \(\sin \theta = \frac { 3 } { 5 }\). The object is in limiting equilibrium and the coefficient of friction at \(A\) is \(\mu\).
2. Find the numerical value of \(\mu\).
3. Given that the resultant force on the object at \(A\) is \(W \sqrt { } ( 65 )\), show that \(k = 5\).

A thin uniform rod \(A B\) has mass \(2 m\) and length \(3 a\). Two identical uniform discs each have mass \(\frac { 1 } { 2 } m\) and radius \(a\). The centre of one of the discs is rigidly attached to the end \(A\) of the rod and the centre of the other disc is rigidly attached to the end \(B\) of the rod. The plane of each disc is perpendicular to the rod \(A B\). A second thin uniform rod \(O C\) has mass \(m\) and length \(2 a\). The end \(C\) of this rod is rigidly attached to the mid-point of \(A B\), with \(O C\) perpendicular to \(A B\) (see diagram). The object consisting of the two discs and two rods is free to rotate about a horizontal axis \(l\), through \(O\), which is perpendicular to both rods.

1. Show that the moment of inertia of one of the discs about \(l\) is \(\frac { 13 } { 4 } m a ^ { 2 }\).
2. Show that the moment of inertia of the object about \(l\) is \(\frac { 52 } { 3 } m a ^ { 2 }\). When the object is suspended from \(O\) and is hanging in equilibrium, the point \(C\) is given a speed of \(\sqrt { } ( 2 a g )\) in the direction parallel to \(A B\). In the subsequent motion, the angle through which \(O C\) has turned before the object comes to instantaneous rest is \(\theta\).
3. Show that \(\cos \theta = \frac { 8 } { 21 }\).