WJEC Further Unit 6 (Further Unit 6) 2023 June

1. The diagram shows a uniform rod \(A B\), of length 8 m and mass 23 kg , in limiting equilibrium with its end \(A\) on rough horizontal ground and point \(C\) resting against a smooth fixed cylinder. The rod is inclined at an angle of \(30 ^ { \circ }\) to the ground.
   \includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-3_240_869_603_598}

The coefficient of friction between the ground and the rod is \(\frac { 2 } { 3 }\).

1. Calculate the magnitude of the normal reaction at \(C\) and the magnitude of the normal reaction to the ground at \(A\).
2. Find the length \(A C\).
3. Suppose instead that the rod is non-uniform with its centre of mass closer to \(A\) than to \(B\). Without carrying out any further calculations, state whether or not this will affect your answers in part (a). Give a reason for your answer.

2. You are given that the centre of mass of a uniform solid cone of height \(h\) and base radius \(r\) is at a height of \(\frac { 1 } { 4 } h\) above its base. The diagram shows a solid conical frustum. It is formed by taking a uniform right circular cone, of base radius \(3 x\) and height \(6 y\), and removing a smaller cone, of base radius \(x\), with the same vertex.
\includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-3_490_903_1937_575} Show that the distance of the centre of mass of the frustum from its base along the axis of symmetry is \(\frac { 18 } { 13 } y\).

3. The vertical motion of a point on the surface of the water in a certain harbour may be modelled as Simple Harmonic Motion about a mean level. The diagram shows that, on a particular day, the depth of water in the harbour at low tide is 2 m and the depth of the water in the harbour at high tide is 10 m . The table below shows the times of high and low tides on this day.
\includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-4_405_912_621_233}

1. Write down the period and amplitude of the motion.
2. Let \(x \mathrm {~m}\) denote the height of water above mean level \(t\) hours after 5a.m. Find an expression for \(x\) in terms of \(t\).
3. The depth of water must be at least 4 m for boats to safely use the harbour. Determine the earliest time, after low tide at 5 a.m., at which boats can safely leave the harbour and hence find the latest possible time of return before the next low tide.
4. Calculate the rate at which the level of water is falling at 2 p.m.

4. The diagram shows three light rods \(A B , B C\) and \(C A\) rigidly joined together so that \(A B C\) is a right-angled triangle with \(A B = 45 \mathrm {~cm} , A C = 28 \mathrm {~cm}\) and \(\widehat { A B } = 90 ^ { \circ }\). The rods support a uniform lamina, of density \(2 m \mathrm {~kg} / \mathrm { cm } ^ { 2 }\), in the shape of a quarter circle \(A D E\) with radius 12 cm and centre at the vertex \(A\). Three particles are attached to \(B C\) : one at \(B\), one at \(C\) and one at \(F\), the midpoint of \(B C\). The masses at \(C , F\) and \(B\) are \(50 m \mathrm {~kg} , 30 m \mathrm {~kg}\) and \(20 m \mathrm {~kg}\) respectively.
\includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-5_604_908_756_575}

1. Calculate the distance of the centre of mass of the system from
   1. \(A C\),
   2. \(A B\).
2. When the system is freely suspended from a point \(P\) on \(A C\), it hangs in equilibrium with \(A B\) vertical. Write down the length \(A P\).
3. When the system is freely suspended from a point \(Q\) on \(A D\), it hangs in equilibrium with \(Q B\) making an angle of \(60 ^ { \circ }\) with the vertical. Find the distance \(A Q\).

5. In this question, \(\mathbf { i }\) and \(\mathbf { j }\) represent unit vectors due east and due north respectively. Two smooth spheres \(P\) and \(Q\), of equal radii, are moving on a smooth horizontal surface. The mass of \(P\) is 2 kg and the mass of \(Q\) is 6 kg . The velocity of \(P\) is \(( 8 \mathbf { i } - 6 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and the velocity of \(Q\) is \(( 4 \mathbf { i } + 10 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). At a particular instant, \(Q\) is positioned 12 m east and 48 m south of \(P\).

1. Prove that \(P\) and \(Q\) will collide. At the instant the spheres collide, the line joining their centres is parallel to the vector \(\mathbf { j }\). Immediately after the collision, sphere \(Q\) has speed \(5 \mathrm {~ms} ^ { - 1 }\).
2. Determine the coefficient of restitution between the spheres and hence calculate the velocity of \(P\) immediately after the collision.
3. Find the magnitude of the impulse required to stop sphere \(P\) after the collision.

6. The diagram on the left shows a train of mass 50 tonnes approaching a buffer at the end of a straight horizontal railway track. The buffer is designed to prevent the train from running off the end of the track. The buffer may be modelled as a light horizontal spring \(A B\), as shown in the diagram on the right, which is fixed at the end \(A\). The train strikes the buffer so that \(P\) makes contact with \(B\) at \(t = 0\) seconds. While \(P\) is in contact with \(B\), an additional resistive force of \(250000 v \mathrm {~N}\) will oppose the motion of the train, where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of the train at time \(t\) seconds. The spring has natural length 1 m and modulus of elasticity 312500 N . At time \(t\) seconds, the compression of the spring is \(x\) metres.
\includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-7_358_1506_824_283}

1. Show that, while \(P\) is in contact with \(B\), \(x\) satisfies the differential equation $$4 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 20 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 25 x = 0$$
2. Given that, when \(P\) first makes contact with \(B\), the speed of the train is \(U \mathrm {~ms} ^ { - 1 }\), find an expression for \(x\) in terms of \(U\) and \(t\).
3. When the train comes to rest, the compression of the buffer is 0.3 m . Determine the speed of the train when it strikes the buffer.
4. State which type of damping is described by the motion of \(P\). Give a reason for your answer.