CAIE Further Paper 3 (Further Paper 3) 2022 November

1 A particle of mass 2 kg is attached to one end of a light inextensible string of length 0.6 m . The other end of the string is attached to a fixed point on a smooth horizontal surface. The particle is moving in a circular path on the surface. The tension in the string is 20 N . Find how many revolutions the particle makes per minute.

2 A light elastic string has natural length \(a\) and modulus of elasticity 4 mg . One end of the string is fixed to a point \(O\) on a smooth horizontal surface. A particle \(P\) of mass \(m\) is attached to the other end of the string. The particle \(P\) is projected along the surface in the direction \(O P\). When the length of the string is \(\frac { 5 } { 4 } a\), the speed of \(P\) is \(v\). When the length of the string is \(\frac { 3 } { 2 } a\), the speed of \(P\) is \(\frac { 1 } { 2 } v\).

1. Find an expression for \(v\) in terms of \(a\) and \(g\).
2. Find, in terms of \(g\), the acceleration of \(P\) when the stretched length of the string is \(\frac { 3 } { 2 } a\).
   \includegraphics[max width=\textwidth, alt={}, center]{7febbd80-4cbb-4b2e-b022-d6a20e7e13aa-04_552_1059_264_502} A smooth cylinder is fixed to a rough horizontal surface with its axis of symmetry horizontal. A uniform rod \(A B\), of length \(4 a\) and weight \(W\), rests against the surface of the cylinder. The end \(A\) of the rod is in contact with the horizontal surface. The vertical plane containing the rod \(A B\) is perpendicular to the axis of the cylinder. The point of contact between the rod and the cylinder is \(C\), where \(A C = 3 a\). The angle between the rod and the horizontal surface is \(\theta\) where \(\tan \theta = \frac { 3 } { 4 }\) (see diagram). The coefficient of friction between the rod and the horizontal surface is \(\frac { 6 } { 7 }\). A particle of weight \(k W\) is attached to the rod at \(B\). The rod is about to slip. The normal reaction between the rod and the cylinder is \(N\).

1. Show that \(\mathrm { N } = \frac { 8 } { 15 } \mathrm {~W} ( 1 + 2 \mathrm { k } )\).
2. Find the value of \(k\).

4 A particle of mass 0.5 kg moves along a horizontal straight line. Its velocity is \(v \mathrm {~ms} ^ { - 1 }\) at time \(t \mathrm {~s}\). The forces acting on the particle are a driving force of magnitude 50 N and a resistance of magnitude \(2 v ^ { 2 } \mathrm {~N}\). The initial velocity of the particle is \(3 \mathrm {~ms} ^ { - 1 }\).

1. Find an expression for \(v\) in terms of \(t\).
2. Deduce the limiting value of \(v\).

5 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 string is held taut with \(O P\) horizontal. The particle \(P\) is projected vertically downwards with speed \(\sqrt { \frac { 1 } { 3 } \mathrm { ag } }\) and starts to move in a vertical circle. \(P\) passes through the lowest point of the circle and reaches the point \(Q\) where \(O Q\) makes an angle \(\theta\) with the downward vertical. At \(Q\) the speed of \(P\) is \(\sqrt { \mathrm { kag } }\) and the tension in the string is \(\frac { 11 } { 6 } \mathrm { mg }\).

1. Find the value of \(k\) and the value of \(\cos \theta\).
   At \(Q\) the particle \(P\) becomes detached from the string.
2. In the subsequent motion, find the greatest height reached by \(P\) above the level of the lowest point of the circle.

6
\includegraphics[max width=\textwidth, alt={}, center]{7febbd80-4cbb-4b2e-b022-d6a20e7e13aa-10_426_1191_267_438} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(k m\) respectively. The two spheres are moving on a horizontal surface with speeds \(u\) and \(\frac { 5 } { 8 } u\) respectively. Immediately before the spheres collide, \(A\) is travelling along the line of centres, and \(B\) 's direction of motion makes an angle \(\alpha\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac { 2 } { 3 }\) and \(\tan \alpha = \frac { 3 } { 4 }\). After the collision, the direction of motion of \(B\) is perpendicular to the line of centres.

1. Find the value of \(k\).
2. Find the loss in the total kinetic energy as a result of the collision.

7 A particle \(P\) is projected with speed \(\mathrm { Vms } ^ { - 1 }\) at an angle \(75 ^ { \circ }\) above the horizontal from a point \(O\) on a horizontal plane. It then moves freely under gravity.

1. Show that the total time of flight, in seconds, is \(\frac { 2 \mathrm {~V} } { \mathrm {~g} } \sin 75 ^ { \circ }\).
   A smooth vertical barrier is now inserted with its lower end on the plane at a distance 15 m from \(O\). The particle is projected as before but now strikes the barrier, rebounds and returns to \(O\). The coefficient of restitution between the barrier and the particle is \(\frac { 3 } { 5 }\).
2. Explain why the total time of flight is unchanged.
3. Find an expression for \(V\) in terms of \(g\).
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