CAIE Further Paper 3 (Further Paper 3) 2024 November

1 A particle of mass 2 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 100 N . The other end of the string is attached to a fixed point \(O\) on a smooth horizontal surface. The particle is moving in a horizontal circle about \(O\) with the string taut and with constant angular speed 5 radians per second. Find the extension of the string.
\includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-02_2717_33_109_2014}

2 A particle \(P\) of mass \(m\) is attached to one end of a light elastic spring of natural length \(a\) and modulus of elasticity 5 mg . The other end of the spring is attached to a fixed point \(O\). The spring hangs vertically with \(P\) below \(O\). The particle \(P\) is pulled down vertically and released from rest when the length of the spring is \(\frac { 3 } { 2 } a\). Find the distance of \(P\) below \(O\) when \(P\) first comes to instantaneous rest.

3
\includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-04_348_828_251_621}
\includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-04_2717_35_110_2012} The diagram shows two identical smooth uniform spheres \(A\) and \(B\) of equal radii and each of mass \(m\). The two spheres are moving on a smooth horizontal surface when they collide with speeds \(2 u\) and \(3 u\) respectively. Immediately before the collision, \(A\) 's direction of motion makes an angle \(\theta\) with the line of centres and \(B\) 's direction of motion is perpendicular to that of \(A\). After the collision, \(B\) moves perpendicular to the line of centres. The coefficient of restitution between the spheres is \(\frac { 1 } { 3 }\).

1. Find the value of \(\tan \theta\).
   \includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-05_2723_33_99_21}
2. Find the total loss of kinetic energy as a result of the collision.
3. Find, in degrees, the angle through which the direction of motion of \(A\) is deflected as a result of the collision.
   \includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-06_776_785_255_680} The end \(A\) of a uniform rod \(A B\) of length \(6 a\) and weight \(W\) is in contact with a rough vertical wall. One end of a light inextensible string of length \(3 a\) is attached to the midpoint \(C\) of the rod. The other end of the string is attached to a point \(D\) on the wall, vertically above \(A\). The rod is in equilibrium when the angle between the rod and the wall is \(\theta\), where \(\tan \theta = \frac { 3 } { 2 }\). A particle of weight \(W\) is attached to the point \(E\) on the rod, where the distance \(A E\) is equal to \(k a ( 3 < k < 6 )\) (see diagram). The rod and the string are in a vertical plane perpendicular to the wall. The coefficient of friction between the rod and the wall is \(\frac { 1 } { 3 }\). The rod is about to slip down the wall.
4. Find the value of \(k\).
   
5. Find, in terms of \(W\), the magnitude of the frictional force between the rod and the wall.

5 A particle \(P\) is projected from a point \(O\) on horizontal ground with speed \(u\) at an angle \(\theta\) above the horizontal, where \(\tan \theta = \frac { 1 } { 3 }\). The particle \(P\) moves freely under gravity and passes through the point with coordinates \(\left( 3 a , \frac { 4 } { 5 } a \right)\) relative to horizontal and vertical axes through \(O\) in the plane of the motion.

1. Use the equation of the trajectory to show that \(u ^ { 2 } = 25 a g\).
   \includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-09_2725_35_99_20} At the instant when \(P\) is moving horizontally, a particle \(Q\) is projected from \(O\) with speed \(V\) at an angle \(\alpha\) above the horizontal. The particles \(P\) and \(Q\) reach the ground at the same point and at the same time.
2. Express \(V ^ { 2 }\) in the form \(k a g\), where \(k\) is a rational number.
   \includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-10_506_803_255_630} 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 with the string taut and the string makes an angle \(\theta\) with the downward vertical through \(O\). The particle \(P\) is projected at right angles to the string with speed \(\frac { 1 } { 3 } \sqrt { 10 a g }\) and begins to move downwards along a circular path. When the string is vertical, it strikes a small smooth peg at the point \(A\) which is vertically below \(O\). The circular path and the point \(A\) are in the same vertical plane. After the string strikes the peg, the particle \(P\) begins to move in a vertical circle with centre \(A\). When the string makes an angle \(\theta\) with the upward vertical through \(A\) the string becomes slack (see diagram). The distance of \(A\) below \(O\) is \(\frac { 5 } { 9 } a\).
3. Find the value of \(\cos \theta\).
   \includegraphics[max width=\textwidth, alt={}, center]{123017e8-8536-4716-aa01-5e9367770575-11_2725_35_99_20}
4. Find the ratio of the tensions in the string immediately before and immediately after it strikes the peg.
   \(7 \quad\) A particle \(P\) of mass \(m \mathrm {~kg}\) is held at rest at a point \(O\) and released so that it moves vertically under gravity against a resistive force of magnitude \(0.1 m v ^ { 2 } \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\).
5. Find an expression for \(v\) in terms of \(t\).
   The displacement of \(P\) from \(O\) at time \(t \mathrm {~s}\) is \(x \mathrm {~m}\).
6. Find an expression for \(v ^ { 2 }\) in terms of \(x\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.
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