CAIE Further Paper 3 (Further Paper 3) 2020 June

1 A particle \(P\) is projected with speed \(u\) at an angle of \(30 ^ { \circ }\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The particle reaches its greatest height at time \(T\) after projection. Find, in terms of \(u\), the speed of \(P\) at time \(\frac { 2 } { 3 } T\) after projection.
\includegraphics[max width=\textwidth, alt={}, center]{7251b13f-1fae-4138-80ea-e6b8091c94ab-04_362_750_258_653} A light inextensible string of length \(a\) is threaded through a fixed smooth ring \(R\). One end of the string is attached to a particle \(A\) of mass \(3 m\). The other end of the string is attached to a particle \(B\) of mass \(m\). The particle \(A\) hangs in equilibrium at a distance \(x\) vertically below the ring. The angle between \(A R\) and \(B R\) is \(\theta\) (see diagram). The particle \(B\) moves in a horizontal circle with constant angular speed \(2 \sqrt { \frac { \mathrm {~g} } { \mathrm { a } } }\). Show that \(\cos \theta = \frac { 1 } { 3 }\) and find \(x\) in terms of \(a\).

3 One end of a light elastic spring, of natural length \(a\) and modulus of elasticity 5 mg , is attached to a fixed point \(A\). The other end of the spring is attached to a particle \(P\) of mass \(m\). The spring hangs with \(P\) vertically below \(A\). The particle \(P\) is released from rest in the position where the extension of the spring is \(\frac { 1 } { 2 } a\).

1. Show that the initial acceleration of \(P\) is \(\frac { 3 } { 2 } g\) upwards.
2. Find the speed of \(P\) when the spring first returns to its natural length.
   \includegraphics[max width=\textwidth, alt={}, center]{7251b13f-1fae-4138-80ea-e6b8091c94ab-08_581_659_267_708} A uniform square lamina \(A B C D\) has sides of length 10 cm . The point \(E\) is on \(B C\) with \(E C = 7.5 \mathrm {~cm}\), and the point \(F\) is on \(D C\) with \(\mathrm { CF } = \mathrm { xcm }\). The triangle \(E F C\) is removed from \(A B C D\) (see diagram). The centre of mass of the resulting shape \(A B E F D\) is a distance \(\bar { x } \mathrm {~cm}\) from \(C B\) and a distance \(\bar { y } \mathrm {~cm}\) from CD.