CAIE Further Paper 3 (Further Paper 3) 2023 June

1 One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(3 m g\), is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(m\). The string hangs with \(P\) vertically below \(O\). The particle \(P\) is pulled vertically downwards so that the extension of the string is \(2 a\). The particle \(P\) is then released from rest.

1. Find the speed of \(P\) when it is at a distance \(\frac { 3 } { 4 } a\) below \(O\).
2. Find the initial acceleration of \(P\) when it is released from rest.
   \includegraphics[max width=\textwidth, alt={}, center]{3b50dc98-781e-4399-8165-ad5e3065df4b-03_741_473_269_836} A particle \(P\) of mass \(m\) is moving with speed \(u\) on a fixed smooth horizontal surface. It collides at an angle \(\alpha\) with a fixed smooth vertical barrier. After the collision, \(P\) moves at an angle \(\theta\) with the barrier, where \(\tan \theta = \frac { 1 } { 2 }\) (see diagram). The coefficient of restitution between \(P\) and the barrier is \(e\). The particle \(P\) loses 20\% of its kinetic energy as a result of the collision. Find the value of \(e\).

3 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 \(P\) is held at the point \(A\), where \(O A\) makes an angle \(\theta\) with the downward vertical through \(O\), and with the string taut. The particle \(P\) is projected perpendicular to \(O A\) in an upwards direction with speed \(u\). It then starts to move along a circular path in a vertical plane. The string goes slack when \(P\) is at \(B\), where angle \(A O B\) is \(90 ^ { \circ }\) and the speed of \(P\) is \(\sqrt { \frac { 4 } { 5 } \mathrm { ag } }\).

1. Find the value of \(\sin \theta\).
2. Find, in terms of \(m\) and \(g\), the tension in the string when \(P\) is at \(A\).
   \includegraphics[max width=\textwidth, alt={}, center]{3b50dc98-781e-4399-8165-ad5e3065df4b-06_846_767_258_689} An object is formed from a solid hemisphere, of radius \(2 a\), and a solid cylinder, of radius \(a\) and height \(d\). The hemisphere and the cylinder are made of the same material. The cylinder is attached to the plane face of the hemisphere. The line \(O C\) forms a diameter of the base of the cylinder, where \(C\) is the centre of the plane face of the hemisphere and \(O\) is common to both circumferences (see diagram). Relative to axes through \(O\), parallel and perpendicular to \(O C\) as shown, the centre of mass of the object is ( \(\mathrm { x } , \mathrm { y }\) ).

1. Show that \(\mathrm { x } = \frac { 32 \mathrm { a } ^ { 2 } + 3 \mathrm { ad } } { 16 \mathrm { a } + 3 \mathrm {~d} }\) and find an expression, in terms of \(a\) and \(d\), for \(\bar { y }\).
   The object is placed on a rough plane which is inclined to the horizontal at an angle \(\theta\) where \(\sin \theta = \frac { 1 } { 6 }\). The object is in equilibrium with \(C O\) horizontal, where \(C O\) lies in a vertical plane through a line of greatest slope.
2. Find \(d\) in terms of \(a\).

5 A light elastic string of natural length \(a\) and modulus of elasticity \(\lambda \mathrm { mg }\) has one end attached to a fixed point \(O\) on a smooth horizontal surface. When a particle of mass \(m\) is attached to the free end of the string, it moves with speed \(v\) in a horizontal circle with centre \(O\) and radius \(x\). When, instead, a particle of mass \(2 m\) is attached to the free end of the string, this particle moves with speed \(\frac { 1 } { 2 } v\) in a horizontal circle with centre \(O\) and radius \(\frac { 3 } { 4 } x\).

1. Find \(x\) in terms of \(a\).
2. Given that \(\mathrm { V } = \sqrt { 12 \mathrm { ag } }\), find the value of \(\lambda\).

6 A particle \(P\) moving in a straight line has displacement \(x \mathrm {~m}\) from a fixed point \(O\) on the line and velocity \(v \mathrm {~ms} ^ { - 1 }\) at time \(t \mathrm {~s}\). The acceleration of \(P\), in \(\mathrm { ms } ^ { - 2 }\), is given by \(6 v \sqrt { v + 9 }\). When \(t = 0 , x = 2\) and \(v = 72\).

1. Find an expression for \(v\) in terms of \(x\).
2. Find an expression for \(x\) in terms of \(t\).

7 At time \(t \mathrm {~s}\), a particle \(P\) is projected with speed \(40 \mathrm {~ms} ^ { - 1 }\) at an angle \(\theta\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The greatest height achieved by \(P\) during its flight is \(H \mathrm {~m}\) and the corresponding time is \(T \mathrm {~s}\).

1. Obtain expressions for \(H\) and \(T\) in terms of \(\theta\).
   During the time between \(t = T\) and \(t = 3 , P\) descends a distance \(\frac { 1 } { 4 } H\).
2. Find the value of \(\theta\).
3. Find the speed of \(P\) when \(t = 3\).
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