CAIE Further Paper 3 (Further Paper 3) 2022 June

1 A uniform lamina \(O A B C\) is a trapezium whose vertices can be represented by coordinates in the \(x - y\) plane. The coordinates of the vertices are \(O ( 0,0 ) , A ( 15,0 ) , B ( 9,4 )\) and \(C ( 3,4 )\). Find the \(x\)-coordinate of the centre of mass of the lamina.

2 A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity \(\frac { 4 } { 3 } \mathrm { mg }\). The other end of the string is attached to a fixed point \(O\) on a rough horizontal surface. The particle is at rest on the surface with the string at its natural length. The coefficient of friction between \(P\) and the surface is \(\frac { 1 } { 3 }\). The particle is projected along the surface in the direction \(O P\) with a speed of \(\frac { 1 } { 2 } \sqrt { \mathrm { ga } }\). Find the greatest extension of the string during the subsequent motion.

3 A particle \(P\) is projected with speed \(25 \mathrm {~ms} ^ { - 1 }\) at an angle \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. After 2 s the speed of \(P\) is \(15 \mathrm {~ms} ^ { - 1 }\).

1. Find the value of \(\sin \theta\).
2. Find the range of the flight.

4 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 and is held with the string taut at the point \(A\). At \(A\) the string makes an angle \(\theta\) with the upward vertical through \(O\). The particle is projected perpendicular to the string in a downward direction from \(A\) with a speed \(u\). It moves along a circular path in the vertical plane. When the string makes an angle \(\alpha\) with the downward vertical through \(O\), the speed of the particle is \(2 u\) and the magnitude of the tension in the string is 10 times its magnitude at \(A\). It is given that \(\mathrm { u } = \sqrt { \frac { 2 } { 3 } \mathrm { ga } }\).

1. Find, in terms of \(m\) and \(g\), the magnitude of the tension in the string at \(A\).
2. Find the value of \(\cos \alpha\).

5 A particle \(P\) of mass 4 kg is moving in a horizontal straight line. At time \(t\) s the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\) and the displacement of \(P\) from a fixed point \(O\) on the line is \(x \mathrm {~m}\). The only force acting on \(P\) is a resistive force of magnitude \(\left( 4 \mathrm { e } ^ { - x } + 12 \right) \mathrm { e } ^ { - x } \mathrm {~N}\). When \(\mathrm { t } = 0 , \mathrm { x } = 0\) and \(v = 4\).

1. Show by integration that \(\mathrm { v } = \frac { 1 + 3 \mathrm { e } ^ { \mathrm { x } } } { \mathrm { e } ^ { \mathrm { x } } }\).
2. Find an expression for \(x\) in terms of \(t\).
   \includegraphics[max width=\textwidth, alt={}, center]{ad8b126c-d739-4e2a-8ce3-7811a61f5876-10_510_889_269_580}
   \(A B\) and \(B C\) are two fixed smooth vertical barriers on a smooth horizontal surface, with angle \(\mathrm { ABC } = 60 ^ { \circ }\). A particle of mass \(m\) is moving with speed \(u\) on the surface. The particle strikes \(A B\) at an angle \(\theta\) with \(A B\). It then strikes \(B C\) and rebounds at an angle \(\beta\) with \(B C\) (see diagram). The coefficient of restitution between the particle and each barrier is \(e\) and \(\tan \theta = 2\). The kinetic energy of the particle after the first collision is \(40 \%\) of its kinetic energy before the first collision.
3. Find the value of \(e\).
4. Find the size of angle \(\beta\).
   \includegraphics[max width=\textwidth, alt={}, center]{ad8b126c-d739-4e2a-8ce3-7811a61f5876-12_965_1059_267_502} A uniform cylinder with a rough surface and of radius \(a\) is fixed with its axis horizontal. Two identical uniform rods \(A B\) and \(B C\), each of weight \(W\) and length \(2 a\), are rigidly joined at \(B\) with \(A B\) perpendicular to \(B C\). The rods rest on the cylinder in a vertical plane perpendicular to the axis of the cylinder with \(A B\) at an angle \(\theta\) to the horizontal. \(D\) and \(E\) are the midpoints of \(A B\) and \(B C\) respectively and also the points of contact of the rods with the cylinder (see diagram). The rods are about to slip in a clockwise direction. The coefficient of friction between each rod and the cylinder is \(\mu\). The normal reaction between \(A B\) and the cylinder is \(R\) and the normal reaction between \(B C\) and the cylinder is \(N\).
5. Find the ratio \(R : N\) in terms of \(\mu\).
6. Given that \(\mu = \frac { 1 } { 3 }\), find the value of \(\tan \theta\).
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