A Level Further Mathematics 9231 Paper 33, May/June 2025

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 $O$. The particle moves in a horizontal circle with constant angular speed $\omega$ and with the string inclined at an angle of $\theta$ to the downward vertical. Given that $\tan \theta =\frac{4}{3}$, find $\omega$ in terms of $a$ and $g$.

A particle $P$ of mass $m$ is attached to one end of a light elastic string of natural length $a$ and modulus of elasticity $mg$. The other end of the string is attached to a fixed point $O$ on a rough plane inclined at an angle of ${30}^{\circ }$ to the horizontal. The particle $P$ is held at rest at point $O$ before being released. The frictional force acting on $P$ as it slides down the plane is $\frac{11}{30}mg$.

A ball of mass $m\text{ kg}$ is projected vertically upwards with initial speed $U{\text{ ms}}^{-1}$ and moves under gravity. At time $t\text{ s}$ after projection, the ball has travelled a distance $x\text{ m}$ and its speed is $v{\text{ ms}}^{-1}$. There is a resistive force of magnitude $mk{v}^2\text{ N}$, where $k$ is a positive constant.

An object consists of a uniform lamina with a particle attached. The uniform lamina $ABCEFD$ of mass $m$ is formed from a rectangle $ABCD$ and an isosceles triangle $CEF$, where $F$ is the midpoint of $CD$. The rectangle has sides $AB=2a$ and $AD=a$. The triangle $CEF$ has base $a$ and height $2a$. The particle of mass $km$ is attached to the lamina at $E$. The object rests in a vertical plane with its edge $AD$ on horizontal ground (see diagram). Given that the object is on the point of toppling in its vertical plane about the vertex $D$, find the value of $k$.

A hollow cylinder of radius $r$ is fixed with its axis horizontal. Points $A$, $B$ and $O$ are in the same vertical plane perpendicular to the axis of the cylinder, with $A$ and $B$ on the smooth inner surface and $O$ on the axis. $OA$ and $OB$ make angles ${90}^{\circ }$ and $\alpha$ respectively with the upward vertical through $O$, with $A$ and $B$ on opposite sides of the vertical. A particle of mass $m$ is projected vertically downwards from point $A$ with speed $\sqrt{\frac{3}{2}rg}$ and moves in a vertical circle inside the cylinder (see diagram). The particle loses contact with the cylinder at point $B$.

Two identical uniform smooth spheres $A$ and $B$, each with mass $m$, are moving on a horizontal surface with speeds $2u$ and $u$ respectively when they collide. Immediately before the collision, the spheres are moving parallel to each other in opposite directions such that their directions of motion each make an angle $\theta$ with the line of centres (see diagram). As a result of the collision, $B$ moves in a direction which is perpendicular to its initial direction of motion. The coefficient of restitution between the spheres is $e$.

A particle $P$ is projected from a point $O$ with speed $U$ at an angle ${45}^{\circ }$ above the horizontal and moves freely under gravity.