CAIE Further Paper 3 (Further Paper 3) 2020 Specimen

A child's toy consists of an iron disc of radius \(r\) and a vertical bead with \(3r\) at rail that is rigidly fixed to the disc so that the toy rocks as it rolls. The circumference of the disc is such that the disc and bead have the same material. Show that the centre of mass of the toy is at a distance \(\frac{27r}{10}\) from the centre of the disc. [4]

A light elastic string of natural length \(a\) and modulus of elasticity \(2mg\). One end of the string is attached to a fixed point \(A\). The other end of the string is attached to a particle of mass \(2m\).

1. Find, in terms of \(a\), the extension of the string when the particle hangs freely in equilibrium below \(A\). [2]
2. The particle is released from rest at \(A\). Find, in terms of \(a\), the distance of the particle below \(A\) when it first comes to instantaneous rest. [6]

A particle \(P\) of mass \(mk\) falls from rest due to gravity. There is a resistance force of magnitude \(mkv^2\) N, where \(v\) ms\(^{-1}\) is the speed of \(P\) after it has fallen a distance \(x\) m and \(k\) is a positive constant.

1. By using \(v \frac{dv}{dx} = \frac{dv}{dt}\) and appropriate differential equation, show that $$v^2 = \frac{g}{k}(1 - e^{-2kx}).$$ [7] It is given that \(k = 0.01\). The speed of \(P\) when \(x = 0.2\) comes to approximately \(v\) ms\(^{-1}\).
2. 1. Find \(V\) correct to 2 decimal places. [1]
   2. Hence find how far \(P\) has fallen when its speed is \(\frac{1}{2}V\) ms\(^{-1}\). [2]

\includegraphics{figure_4} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(2m\) respectively. Sphere \(B\) is at rest on a smooth horizontal surface. Sphere \(A\) is moving on the surface with speed \(u\) at an angle of \(30°\) to the line of centres of \(A\) and \(B\) when it collides with \(B\) (see diagram). The coefficient of restitution between the spheres is \(e\).

1. Show that the speed of \(B\) after the collision is \(\frac{\sqrt{3}}{6}u(1 + e)\) and find the speed of \(A\) after the collision. [6]
2. Given that \(e = \frac{1}{2}\), find the loss of kinetic energy as a result of the collision. [3]

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 \(O\).

1. \includegraphics{figure_5a} The particle \(P\) moves in a horizontal circle with a constant angular speed \(\omega\) with the string inclined at \(60°\) to the downward vertical through \(O\) (see diagram). Show that \(\omega^2 = \frac{2g}{a}\). [4]
2. The particle now hangs at rest and is then projected horizontally so that it begins to move in a vertical circle with centre \(O\). When the string makes an angle \(\theta\) with the downward vertical through \(O\), the angular speed of \(P\) is \(\sqrt{\frac{2g}{a}}\). The string first goes slack when \(OP\) makes an angle \(\theta\) with the upward vertical through \(O\). Find the value of \(\cos \theta\). [6]

A particle \(P\) is projected with speed \(u\) at an angle \(\alpha\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of \(P\) from \(O\) at a time \(t\) are denoted by \(x\) and \(y\) respectively.

1. Derive the equation of the trajectory of \(P\) in the form $$y = x \tan \alpha - \frac{gx^2}{2u^2} \sec^2 \alpha.$$ [3]
2. The greatest height of \(P\) above the plane is denoted by \(H\). When \(P\) is at a height of \(\frac{3}{4}H\), it is travelling at a horizontal distance \(d\). Given that \(\tan \alpha = 3\) and in terms of \(H\), the two possible values of \(d\). [6]