CAIE Further Paper 3 (Further Paper 3) 2023 November

2 A ball of mass 2 kg is projected vertically downwards with speed \(5 \mathrm {~ms} ^ { - 1 }\) through a liquid. At time \(t \mathrm {~s}\) after projection, the velocity of the ball is \(v \mathrm {~ms} ^ { - 1 }\) and its displacement from its starting point is \(x \mathrm {~m}\). The forces acting on the ball are its weight and a resistive force of magnitude \(0.2 v ^ { 2 } \mathrm {~N}\).

1. Find an expression for \(v\) in terms of \(t\).
2. Deduce what happens to \(v\) for large values of \(t\).
   \includegraphics[max width=\textwidth, alt={}, center]{6964a0b0-8fc8-4fa8-b3fe-f51dafdaaeec-06_803_652_251_703} A uniform square lamina of side \(2 a\) and weight \(W\) is suspended from a light inextensible string attached to the midpoint \(E\) of the side \(A B\). The other end of the string is attached to a fixed point \(P\) on a rough vertical wall. The vertex \(B\) of the lamina is in contact with the wall. The string \(E P\) is perpendicular to the side \(A B\) and makes an angle \(\theta\) with the wall (see diagram). The string and the lamina are in a vertical plane perpendicular to the wall. The coefficient of friction between the wall and the lamina is \(\frac { 1 } { 2 }\). Given that the vertex \(B\) is about to slip up the wall, find the value of \(\tan \theta\).
   \includegraphics[max width=\textwidth, alt={}, center]{6964a0b0-8fc8-4fa8-b3fe-f51dafdaaeec-08_581_576_269_731} A light elastic string has natural length \(8 a\) and modulus of elasticity \(5 m g\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The ends of the string are attached to points \(A\) and \(B\) which are a distance \(12 a\) apart on a smooth horizontal table. The particle \(P\) is held on the table so that \(A P = B P = L\) (see diagram). The particle \(P\) is released from rest. When \(P\) is at the midpoint of \(A B\) it has speed \(\sqrt { 80 a g }\).
3. Find \(L\) in terms of \(a\).
4. Find the initial acceleration of \(P\) in terms of \(g\).

5 A particle \(P\) is projected with speed \(u \mathrm {~ms} ^ { - 1 }\) at an angle \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. During its flight \(P\) passes through the point which is a horizontal distance \(3 a\) from \(O\) and a vertical distance \(\frac { 3 } { 8 } a\) above the horizontal plane. It is given that \(\tan \theta = \frac { 1 } { 3 }\).

1. Show that \(\mathrm { u } ^ { 2 } = 8 \mathrm { ag }\).
   A particle \(Q\) is projected with speed \(V \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal from \(O\) at the instant when \(P\) is at its highest point. Particles \(P\) and \(Q\) both land at the same point on the horizontal plane at the same time.
2. Find \(V\) in terms of \(a\) and \(g\).

6 A particle \(P\) of mass \(m\) is attached to one end of a light inextensible rod of length \(3 a\). An identical particle \(Q\) is attached to the other end of the rod. The rod is smoothly pivoted at a point \(O\) on the rod, where \(\mathrm { OQ } = \mathrm { x }\). The system, of rod and particles, rotates about \(O\) in a vertical plane. At an instant when the rod is vertical, with \(P\) above \(Q\), the particle \(P\) is moving horizontally with speed \(u\). When the rod has turned through an angle of \(60 ^ { \circ }\) from the vertical, the speed of \(P\) is \(2 \sqrt { \mathrm { ag } }\), and the tensions in the two parts of the rod, \(O P\) and \(O Q\), have equal magnitudes.

1. Show that the speed of \(Q\) when the rod has turned through an angle of \(60 ^ { \circ }\) from the vertical is \(\frac { 2 x } { 3 a - x } \sqrt { a g }\).
2. Find \(x\) in terms of \(a\).
3. Find \(u\) in terms of \(a\) and \(g\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.