CAIE Further Paper 3 (Further Paper 3) 2021 November

1 One end of a light elastic string, of natural length \(a\) and modulus of elasticity 3 mg , is attached to a fixed point \(O\) on a smooth horizontal plane. A particle \(P\) of mass \(m\) is attached to the other end of the string and moves in a horizontal circle with centre \(O\). The speed of \(P\) is \(\sqrt { \frac { 4 } { 3 } \mathrm { ga } \text {. } }\) Find the extension of the string.

2 A particle \(P\) of mass \(m \mathrm {~kg}\) moves along a horizontal straight line with acceleration \(a \mathrm {~ms} ^ { - 2 }\) given by $$a = \frac { v \left( 1 - 2 t ^ { 2 } \right) } { t }$$ where \(v \mathrm {~ms} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\).

1. Find an expression for \(v\) in terms of \(t\) and an arbitrary constant.
2. Given that \(a = 5\) when \(t = 1\), find an expression, in terms of \(m\) and \(t\), for the horizontal force acting on \(P\) at time \(t\).

3 A light elastic string has natural length \(a\) and modulus of elasticity 12 mg . One end of the string is attached to a fixed point \(O\). The other end of the string is attached to a particle of mass \(m\). The particle hangs in equilibrium vertically below \(O\). The particle is pulled vertically down and released from rest with the extension of the string equal to \(e\), where \(\mathrm { e } > \frac { 1 } { 3 } \mathrm { a }\). In the subsequent motion the particle has speed \(\sqrt { 2 \mathrm { ga } }\) when it has ascended a distance \(\frac { 1 } { 3 } a\). Find \(e\) in terms of \(a\).
\includegraphics[max width=\textwidth, alt={}, center]{e4926d36-7246-4cde-b466-44ecc4c30a61-06_488_496_269_781} A uniform lamina \(A E C F\) is formed by removing two identical triangles \(B C E\) and \(C D F\) from a square lamina \(A B C D\). The square has side \(3 a\) and \(E B = D F = h\) (see diagram).

1. Find the distance of the centre of mass of the lamina \(A E C F\) from \(A D\) and from \(A B\), giving your answers in terms of \(a\) and \(h\).
   The lamina \(A E C F\) is placed vertically on its edge \(A E\) on a horizontal plane.
2. Find, in terms of \(a\), the set of values of \(h\) for which the lamina remains in equilibrium.

5 A particle \(P\) is projected from a point \(O\) on a horizontal plane and moves freely under gravity. Its initial speed is \(u \mathrm {~ms} ^ { - 1 }\) and its angle of projection is \(\sin ^ { - 1 } \left( \frac { 4 } { 5 } \right)\) above the horizontal. At time 8 s after projection, \(P\) is at the point \(A\). At time 32 s after projection, \(P\) is at the point \(B\). The direction of motion of \(P\) at \(B\) is perpendicular to its direction of motion at \(A\). Find the value of \(u\).

6 A particle \(P\), of mass \(m\), is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) moves in complete vertical circles about \(O\) with the string taut. The points \(A\) and \(B\) are on the path of \(P\) with \(A B\) a diameter of the circle. \(O A\) makes an angle \(\theta\) with the downward vertical through \(O\) and \(O B\) makes an angle \(\theta\) with the upward vertical through \(O\). The speed of \(P\) when it is at \(A\) is \(\sqrt { 5 a g }\). The ratio of the tension in the string when \(P\) is at \(A\) to the tension in the string when \(P\) is at \(B\) is \(9 : 5\).

1. Find the value of \(\cos \theta\).
2. Find, in terms of \(a\) and \(g\), the greatest speed of \(P\) during its motion.
   \includegraphics[max width=\textwidth, alt={}, center]{e4926d36-7246-4cde-b466-44ecc4c30a61-12_613_718_251_676} The smooth vertical walls \(A B\) and \(C B\) are at right angles to each other. A particle \(P\) is moving with speed \(u\) on a smooth horizontal floor and strikes the wall \(C B\) at an angle \(\alpha\). It rebounds at an angle \(\beta\) to the wall \(C B\). The particle then strikes the wall \(A B\) and rebounds at an angle \(\gamma\) to that wall (see diagram). The coefficient of restitution between each wall and \(P\) is \(e\).
3. Show that \(\tan \beta = e \tan \alpha\).
4. Express \(\gamma\) in terms of \(\alpha\) and explain what this result means about the final direction of motion of \(P\).
   As a result of the two impacts the particle loses \(\frac { 8 } { 9 }\) of its initial kinetic energy.
5. Given that \(\alpha + \beta = 90 ^ { \circ }\), find the value of \(e\) and the value of \(\tan \alpha\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.