A Level Further Mathematics 9231 Paper 31, May/June 2024

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 $\frac{1}{2}$. Find the value of $\tan \theta$.

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 $\frac{3}{2}a$ below $M$, the midpoint of $AB$.

At time $t=0\text{ seconds}$, a particle $P$ is projected with speed $u\text{ m}{\text{s}}^{-1}$ at an angle ${60}^{\circ }$ 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 $u$.

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 $\tan \alpha =\frac{1}{2}$. 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 $\mu$.

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 $\frac{1}{5}$. When the turntable is made to rotate with angular speed $\frac{2}{5}\sqrt{\frac{g}{a}}$, the system is in limiting equilibrium.

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

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 $O$.