AQA Further AS Paper 2 Mechanics (Further AS Paper 2 Mechanics) 2019 June

A turntable rotates at a constant speed of \(33\frac{1}{3}\) revolutions per minute. Find the angular speed in radians per second. Circle your answer. [1 mark] \(\frac{5\pi}{9}\) \quad \(\frac{10\pi}{9}\) \quad \(\frac{5\pi}{3}\) \quad \(\frac{20\pi}{9}\)

The graph shows the resistance force experienced by a cyclist over the first 20 metres of a bicycle ride. \includegraphics{figure_2} Find the work done by the resistance force over the 20 metres of the bicycle ride. Circle your answer. [1 mark] 1600 J \quad 3000 J \quad 3200 J \quad 4000 J

A formula for the elastic potential energy, \(E\), stored in a stretched spring is given by $$E = \frac{kx^2}{2}$$ where \(x\) is the extension of the spring and \(k\) is a constant. Use dimensional analysis to find the dimensions of \(k\). [3 marks]

In this question use \(g = 9.8\,\text{m}\,\text{s}^{-2}\) A ride in a fairground consists of a hollow vertical cylinder of radius 4.6 metres with a horizontal floor. Stephi, who has mass 50 kilograms, stands inside the cylinder with her back against the curved surface. The cylinder begins to rotate about a vertical axis through the centre of the cylinder. When the cylinder is rotating at a constant angular speed of \(\omega\) radians per second, the magnitude of the normal reaction between Stephi and the curved surface is 980 newtons. The floor is lowered and Stephi remains against the curved surface with her feet above the floor, as shown in the diagram. \includegraphics{figure_4}

1. Explain, with the aid of a force diagram, why the magnitude of the frictional force acting on Stephi is 490 newtons. [2 marks]
2. Find \(\omega\) [3 marks]
3. State one modelling assumption that you have used in this question. Explain the effect of this assumption. [2 marks]

A car of mass 1000 kg has a maximum speed of \(40\,\text{m}\,\text{s}^{-1}\) when travelling on a straight horizontal race track. The maximum power output of the car's engine is 48 kW The total resistance force experienced by the car can be modelled as being proportional to the car's speed. Find the maximum possible acceleration of the car when it is travelling at \(25\,\text{m}\,\text{s}^{-1}\) on the straight horizontal race track. Fully justify your answer. [7 marks]

In this question use \(g = 9.8\,\text{m}\,\text{s}^{-2}\) Martin, who is of mass 40 kg, is using a slide. The slide is made of two straight sections \(AB\) and \(BC\). The section \(AB\) has length 15 metres and is at an angle of \(50°\) to the horizontal. The section \(BC\) has length 2 metres and is horizontal. \includegraphics{figure_6} Martin pushes himself from \(A\) down the slide with initial speed \(1\,\text{m}\,\text{s}^{-1}\) He reaches \(B\) with speed \(5\,\text{m}\,\text{s}^{-1}\) Model Martin as a particle.

1. Find the energy lost as Martin slides from \(A\) to \(B\). [4 marks]
2. Assume that a resistance force of constant magnitude acts on Martin while he is moving on the slide.
   1. Show that the magnitude of this resistance force is approximately 270 N [2 marks]
   2. Determine if Martin reaches the point \(C\). [3 marks]

Two smooth spheres, \(P\) and \(Q\), of equal radius are free to move on a smooth horizontal surface. The masses of \(P\) and \(Q\) are \(3m\) and \(m\) respectively. \(P\) is set in motion with speed \(u\) directly towards \(Q\), which is initially at rest. \(P\) subsequently collides with \(Q\). \includegraphics{figure_7} Immediately after the collision, \(P\) moves with speed \(v\) and \(Q\) moves with speed \(w\). The coefficient of restitution between the spheres is \(e\).

1. 1. Show that $$v = \frac{u(3-e)}{4}$$ [4 marks]
   2. Find \(w\), in terms of \(e\) and \(u\), simplifying your answer. [2 marks]
2. Deduce that $$\frac{u}{2} \leq v \leq \frac{3u}{4}$$ [2 marks]
3. 1. Find, in terms of \(m\) and \(u\), the maximum magnitude of the impulse that \(P\) exerts on \(Q\). [3 marks]
   2. Describe the impulse that \(Q\) exerts on \(P\). [1 mark]