CAIE Further Paper 3 (Further Paper 3) 2024 June

Two smooth uniform spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(5m\) respectively. Sphere \(A\) is moving on a smooth horizontal surface with speed \(u\) when it collides with sphere \(B\) which is at rest on the surface. Immediately before the collision, \(A\)'s direction of motion makes an angle of \(\theta\) with the line of centres. After the collision, the kinetic energies of \(A\) and \(B\) are equal. The coefficient of restitution between the spheres is \(\frac{1}{3}\). \includegraphics{figure_1} Find the value of \(\tan\theta\). [6]

The points \(A\) and \(B\) are at the same horizontal level a distance \(4a\) apart. The ends of a light elastic string, of natural length \(4a\) and modulus of elasticity \(\lambda\), are attached to \(A\) and \(B\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The system is in equilibrium with \(P\) at a distance \(\frac{5}{8}a\) below \(M\), the midpoint of \(AB\).

1. Find \(\lambda\) in terms of \(m\) and \(g\). [3]

The particle \(P\) is pulled down vertically and released from rest at a distance \(\frac{8}{5}a\) below \(M\).

1. Find, in terms of \(a\) and \(g\), the speed of \(P\) as it passes through \(M\) in the subsequent motion. [4]

At time \(t = 0\) seconds, a particle \(P\) is projected with speed \(u\) m s\(^{-1}\) at an angle \(60°\) above the horizontal from a point \(O\). In the subsequent motion \(P\) moves freely under gravity. The direction of motion of \(P\) when \(t = 5\) is perpendicular to its direction of motion when \(t = 15\). Find the value of \(u\). [5]

A ring of weight \(W\), with radius \(a\) and centre \(O\), is at rest on a rough surface that is inclined to the horizontal at an angle \(\alpha\) where \(\tan\alpha = \frac{1}{3}\). The plane of the ring is perpendicular to the inclined surface and parallel to a line of greatest slope of the surface. The point \(P\) on the circumference of the ring is such that \(OP\) is parallel to the surface. A light inextensible string is attached to \(P\) and to the point \(Q\), which is on the surface, such that \(PQ\) is horizontal (see diagram). The points \(O\), \(P\) and \(Q\) are in the same vertical plane. The system is in limiting equilibrium and the coefficient of friction between the ring and the surface is \(\mu\). \includegraphics{figure_4}

1. Find, in terms of \(W\), the tension in the string \(PQ\). [4]
2. Find the value of \(\mu\). [3]

Two particles \(A\) and \(B\) of masses \(m\) and \(km\) respectively are connected by a light inextensible string of length \(a\). The particles are placed on a rough horizontal circular turntable with the string taut and lying along a radius of the turntable. Particle \(A\) is at a distance \(a\) from the centre of the turntable and particle \(B\) is at a distance \(2a\) from the centre of the turntable. The coefficient of friction between each particle and the turntable is \(\frac{1}{3}\). When the turntable is made to rotate with angular speed \(\frac{2}{5}\sqrt{\frac{g}{a}}\), the system is in limiting equilibrium.

1. Find the tension in the string, in terms of \(m\) and \(g\). [4]
2. Find the value of \(k\). [3]

A particle \(P\) of mass \(2\) kg moving on a horizontal straight line has displacement \(x\) m from a fixed point \(O\) on the line and velocity \(v\) m s\(^{-1}\) at time \(t\) s. The only horizontal force acting on \(P\) has magnitude \(\frac{1}{10}(2v - 1)^2e^{-t}\) N and acts towards \(O\). When \(t = 0\), \(x = 1\) and \(v = 3\).

1. Find an expression for \(v\) in terms of \(t\). [5]
2. Find an expression for \(x\) in terms of \(t\). [4]

A smooth sphere with centre \(O\) and of radius \(a\) is fixed to a horizontal plane. A particle \(P\) of mass \(m\) is projected horizontally from the highest point of the sphere with speed \(u\), so that it begins to move along the surface of the sphere. The particle \(P\) loses contact with the sphere at the point \(Q\) on the sphere, where \(OQ\) makes an angle \(\theta\) with the upward vertical through \(O\).

1. Show that \(\cos\theta = \frac{u^2 + 2ag}{3ag}\). [4]

It is given that \(\cos\theta = \frac{5}{9}\).

1. Find, in terms of \(a\) and \(g\), an expression for the vertical component of the velocity of \(P\) just before it hits the horizontal plane to which the sphere is fixed. [3]
2. Find an expression for the time taken by \(P\) to fall from \(Q\) to the plane. Give your answer in the form \(k\sqrt{\frac{a}{g}}\), stating the value of \(k\) correct to 3 significant figures. [2]