AQA Further Paper 3 Mechanics (Further Paper 3 Mechanics) 2021 June

A spring of natural length 50 cm and modulus of elasticity \(\lambda\) newtons has an elastic potential energy of 4 J when compressed by 5 cm. Find the value of \(\lambda\) Circle your answer. [1 mark] 8 16 800 1600

A force of magnitude 7 N acts at each end of a rod of length 20 cm, forming a couple. The forces act at right angles to the rod, as shown in the diagram below. \includegraphics{figure_2} Find the magnitude of the resultant moment of the couple. Circle your answer. [1 mark] 1.4 N m 2.8 N m 140 N m 280 N m

A ball has mass 0.4 kg and is hit by a wooden bat. The speed of the ball just before it is hit by the bat is \(6 \text{ m s}^{-1}\) The velocity of the ball immediately after being hit by the bat is perpendicular to its initial velocity. The speed of the ball just after it is hit by the bat is \(8 \text{ m s}^{-1}\) Show that the impulse on the ball has magnitude 4 N s [3 marks]

A spring has stiffness \(k\)

1. Determine the dimensions of \(k\) [1 mark]
2. One end of the spring is attached to a fixed point. A particle of mass \(m\) kg is attached to the other end of the spring. The particle is set into vertical motion and moves up and down, taking \(t\) seconds to complete one oscillation. A possible model for \(t\) is $$t = pm^a g^b k^c$$ where \(p\) is a dimensionless constant and \(g \text{ m s}^{-2}\) is the acceleration due to gravity. Find the values of \(a\), \(b\) and \(c\) for this model to be dimensionally consistent. [3 marks]

A uniform lamina has the shape of the region enclosed by the curve \(y = x^2 + 1\) and the lines \(x = 0\), \(x = 4\) and \(y = 0\) The diagram below shows the lamina. \includegraphics{figure_5}

1. Find the coordinates of the centre of mass of the lamina, giving your answer in exact form. [4 marks]
2. The lamina is suspended from the point where the curve intersects the line \(x = 4\) and hangs in equilibrium. Find the angle between the vertical and the longest straight edge of the lamina, giving your answer correct to the nearest degree. [3 marks]

A ball of mass \(m\) kg is held at rest at a height \(h\) metres above a horizontal surface. The ball is released and bounces on the surface. The coefficient of restitution between the ball and the surface is \(e\) Prove that the kinetic energy lost during the first bounce is given by $$mgh(1 - e^2)$$ [4 marks]

A light string has length 1.5 metres. A small sphere is attached to one end of the string. The other end of the string is attached to a fixed point O A thin horizontal bar is positioned 0.9 metres directly below O The bar is perpendicular to the plane in which the sphere moves. The sphere is released from rest with the string taut and at an angle \(\alpha\) to the downward vertical through O The string becomes slack when the angle between the two sections of the string is 60° Ben draws the diagram below to show the initial position of the sphere, the bar and the path of the sphere. \includegraphics{figure_7}

1. State two reasons why Ben's diagram is not a good representation of the situation. [2 marks]
2. Using your answer to part (a), sketch an improved diagram. [1 mark]
3. Find \(\alpha\), giving your answer to the nearest degree. [6 marks]

In this question use \(g = 9.8 \text{ m s}^{-2}\) A lift is used to raise a crate of mass 250 kg The lift exerts an upward force of magnitude \(P\) newtons on the crate. When the crate is at a height of \(x\) metres above its initial position $$P = k(x + 1)(12 - x) + 2450$$ where \(k\) is a constant. The crate is initially at rest, at the point where \(x = 0\)

1. Show that the work done by the upward force as the crate rises to a height of 12 metres is given by $$29400 + 360k$$ [3 marks]
2. The speed of the crate is \(3 \text{ m s}^{-1}\) when it has risen to a height of 12 metres. Find the speed of the crate when it has risen to a height of 15 metres. [5 marks]
3. Find the height of the crate when its speed becomes zero. [2 marks]
4. Air resistance has been ignored. Explain why this is reasonable in this context. [1 mark]

In this question use \(g = 9.81 \text{ m s}^{-2}\) A conical pendulum is made from an elastic string and a sphere of mass 0.2 kg The string has natural length 1.6 metres and modulus of elasticity 200 N The sphere describes a horizontal circle of radius 0.5 metres at a speed of \(v \text{ m s}^{-1}\) The angle between the elastic string and the vertical is \(\alpha\)

1. Show that $$62.5 - 200 \sin \alpha = 1.962 \tan \alpha$$ [5 marks]
2. Use your calculator to find \(\alpha\) [1 mark]
3. Find the value of \(v\) [4 marks]