CAIE Further Paper 3 (Further Paper 3) 2022 June

2 One end of a light inextensible string of length \(a\) is attached to a fixed point \(O\). A particle of mass \(m\) is attached to the other end of the string. The particle is held at the point \(A\) with the string taut. The angle between \(O A\) and the downward vertical is equal to \(\alpha\), where \(\cos \alpha = \frac { 4 } { 5 }\). The particle is projected from \(A\), perpendicular to the string in an upwards direction, with a speed \(\sqrt { 3 \text { ga } }\). It then moves along a circular path in a vertical plane. The string first goes slack when it makes an angle \(\theta\) with the upward vertical through \(O\). Find the value of \(\cos \theta\).

3 A particle \(P\) is moving in a horizontal straight line. Initially \(P\) is at the point \(O\) on the line and is moving with velocity \(25 \mathrm {~ms} ^ { - 1 }\). At time \(t \mathrm {~s}\) after passing through \(O\), the acceleration of \(P\) is \(\frac { 4000 } { ( 5 t + 4 ) ^ { 3 } } \mathrm {~ms} ^ { - 2 }\) in the direction \(P O\). The displacement of \(P\) from \(O\) at time \(t\) is \(x \mathrm {~m}\). Find an expression for \(x\) in terms of \(t\).
\includegraphics[max width=\textwidth, alt={}, center]{9067c549-00d7-4078-b47d-87b28396e2ab-06_894_809_260_628} An object is composed of a hemispherical shell of radius \(2 a\) attached to a closed hollow circular cylinder of height \(h\) and base radius \(a\). The hemispherical shell and the hollow cylinder are made of the same uniform material. The axes of symmetry of the shell and the cylinder coincide. \(A B\) is a diameter of the lower end of the cylinder (see diagram).

1. Find, in terms of \(a\) and \(h\), an expression for the distance of the centre of mass of the object from \(A B\). [4]
   The object is placed on a rough plane which is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac { 2 } { 3 }\). The object is in equilibrium with \(A B\) in contact with the plane and lying along a line of greatest slope of the plane.
2. Find the set of possible values of \(h\), in terms of \(a\).
   \includegraphics[max width=\textwidth, alt={}, center]{9067c549-00d7-4078-b47d-87b28396e2ab-08_629_1358_269_367} A light inextensible string \(A B\) passes through two small holes \(C\) and \(D\) in a smooth horizontal table where \(A C = 3 a\) and \(D B = a\). A particle of mass \(m\) is attached at the end \(A\) and moves in a horizontal circle with angular velocity \(\omega\). A particle of mass \(\frac { 3 } { 4 } m\) is attached to the end \(B\) and moves in a horizontal circle with angular velocity \(k \omega\). \(A C\) makes an angle \(\theta\) with the downward vertical and \(D B\) makes an angle \(\theta\) with the horizontal (see diagram). Find the value of \(k\).

6 Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(k m\) respectively. The two spheres are on a horizontal surface. Sphere \(A\) is travelling with speed \(u\) towards sphere \(B\) which is at rest. The spheres collide. Immediately before the collision, the direction of motion of \(A\) makes an angle \(\alpha\) with the line of centres. The coefficient of restitution between the spheres is \(\frac { 1 } { 2 }\).

1. Show that the speed of \(B\) after the collision is \(\frac { 3 \mathrm { u } \cos \alpha } { 2 ( 1 + \mathrm { k } ) }\) and find also an expression for the speed of \(A\) along the line of centres after the collision, in terms of \(k , u\) and \(\alpha\).
   After the collision, the kinetic energy of \(A\) is equal to the kinetic energy of \(B\).
2. Given that \(\tan \alpha = \frac { 2 } { 3 }\), find the possible values of \(k\).

7 Particles \(P\) and \(Q\) are projected in the same vertical plane from a point \(O\) at the top of a cliff. The height of the cliff exceeds 50 m . Both particles move freely under gravity. Particle \(P\) is projected with speed \(\frac { 35 } { 2 } \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac { 4 } { 3 }\). Particle \(Q\) is projected with speed \(u \mathrm {~ms} ^ { - 1 }\) at an angle \(\beta\) above the horizontal, where \(\tan \beta = \frac { 1 } { 2 }\). Particle \(Q\) is projected one second after the projection of particle \(P\). The particles collide \(T\) s after the projection of particle \(Q\).

1. Write down expressions, in terms of \(T\), for the horizontal displacements of \(P\) and \(Q\) from \(O\) when they collide and hence show that \(4 \mathrm { uT } = 21 \sqrt { 5 } ( \mathrm {~T} + 1 )\).
2. Find the value of \(T\).
3. Find the horizontal and vertical displacements of the particles from \(O\) when they collide.
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