WJEC Further Unit 3 (Further Unit 3) 2023 June

One end of a light elastic string, of natural length \(0.2\) m and modulus of elasticity \(5g\) N, is attached to a fixed point \(O\). The other end is attached to a particle of mass \(4\) kg. The particle hangs in equilibrium vertically below \(O\).

1. Show that the extension of the string is \(0.16\) m. [2]
2. The particle is pulled down vertically and held at rest so that the extension of the string is \(0.28\) m. The particle is then released. Determine the speed of the particle as it passes through the equilibrium position. [8]

At time \(t = 0\) seconds, a particle \(A\) has position vector \((6\mathbf{i} + 2\mathbf{j} - 8\mathbf{k})\) m relative to a fixed origin \(O\) and is moving with constant velocity \((3\mathbf{i} - \mathbf{j} + 4\mathbf{k})\) ms\(^{-1}\).

1. Write down the position vector of particle \(A\) at time \(t\) seconds and hence find the distance \(OA\) when \(t = 5\). [4]
2. The position vector, \(\mathbf{r}_B\) metres, of another particle \(B\) at time \(t\) seconds is given by $$\mathbf{r}_B = 3\sin\left(\frac{t}{2}\right)\mathbf{i} - 3\cos\left(\frac{t}{2}\right)\mathbf{j} + 5\mathbf{k}.$$
   1. Show that \(B\) is moving with constant speed.
   2. Determine the smallest value of \(t\) such that particles \(A\) and \(B\) are moving perpendicular to each other. [7]

The diagram below shows a hollow cone, of base radius \(5\) m and height \(12\) m, which is fixed with its axis vertical and vertex \(V\) downwards. A particle \(P\), of mass \(M\) kg, moves in a horizontal circle with centre \(C\) on the smooth inner surface of the cone with constant speed \(v = 3\sqrt{g}\) ms\(^{-1}\). \includegraphics{figure_3}

1. Show that the normal reaction of the surface of the cone on the particle is \(\frac{13Mg}{5}\) N. [4]
2. Calculate the length of \(CP\) and hence determine the height of \(C\) above \(V\). [6]

Geraint is a cyclist competing in a race along the Taff Trail. The Taff Trail is a track that runs from Cardiff Bay to Brecon. The chart below shows the altitude (height above sea level) along the route. \includegraphics{figure_4} Geraint starts from rest at Cardiff Bay and has a speed of \(10\) ms\(^{-1}\) when he crosses the finish line in Brecon. Geraint and his bike have a total mass of \(80\) kg. The resistance to motion may be modelled by a constant force of magnitude \(16\) N.

1. Given that \(1440\) kJ of energy is used in overcoming resistances during the race,
   1. find the length of the track,
   2. calculate the work done by Geraint. [8]
2. The steepest section of the track may be modelled as a slope inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{2}{7}\). \includegraphics{figure_4b} Geraint is capable of producing a maximum power of \(250\) W. Find the maximum speed that Geraint can attain whilst travelling on this section of the track. [5]

The diagram below shows two spheres \(A\) and \(B\), of equal radii, moving in the same direction on a smooth horizontal surface. Sphere \(A\), of mass \(3\) kg, is moving with speed \(4\) ms\(^{-1}\) and sphere \(B\), of mass \(2\) kg, is moving with speed \(10\) ms\(^{-1}\). \includegraphics{figure_5} Sphere \(B\) is then given an impulse after which it moves in the opposite direction with speed \(6\) ms\(^{-1}\).

1. Calculate the magnitude and direction of the impulse exerted on \(B\). [3]

Sphere \(B\) continues to move with speed \(6\) ms\(^{-1}\) so that it collides directly with sphere \(A\). The kinetic energy lost due to the collision is \(45\) J.

1. Calculate the speed of \(A\) and the speed of \(B\) immediately after the two spheres collide. State the direction in which each sphere is moving relative to its motion immediately before the collision. [8]

The diagram shows a slide, \(ABC\), at a water park. The shape of the slide may be modelled by two circular arcs, \(AB\) and \(BC\), in the same vertical plane. Arc \(AB\) has radius \(7\) m and subtends an angle \(\alpha\) at its centre \(D\), where \(\cos \alpha = \frac{9}{10}\). Arc \(BC\) has radius \(5\) m and subtends an angle of \(45°\) at its centre, \(O\). The straight line \(DBO\) is vertical. \includegraphics{figure_6} Users of the slide are required to sit in a rubber ring and are released from rest at point \(A\). A girl decides to use the slide. The combined mass of the girl and the rubber ring is \(50\) kg.

1. When the rubber ring is at a point \(P\) on the circular arc \(BC\), its speed is \(v\) ms\(^{-1}\) and \(OP\) makes an angle \(\theta\) with the upward vertical.
   1. Show that \(v^2 = 111.72 - 98\cos\theta\). [4]
   2. Find, in terms of \(\theta\), the reaction between the rubber ring and the slide at \(P\). [4]
   3. Show that, according to this model, the rubber ring loses contact with the slide before reaching \(C\). [3]
   4. In reality, there will be resistive forces opposing the motion of the rubber ring. Explain how this fact will affect your answer to (iii). [1]
2. Show that the rubber ring will remain in contact with the slide along the arc \(AB\). [3]