A Level Further Mathematics 9231 Paper 32, Oct/Nov 2025

A particle $P$ of mass $m$ is attached to two light inextensible strings each of length $l$. The end of one string is attached to a fixed point $A$ and the end of the other string is attached to a fixed point $B$, with $A$ vertically above $B$. Angle $APB$ is a right angle. The particle $P$ rotates in a horizontal circle at a constant angular speed $\omega$ with both strings taut (see diagram). Find the tension in string $AP$ in terms of $m$, $g$, $l$ and $\omega$.

One end of a light elastic string of natural length $a$ and modulus of elasticity $2mg$ is attached to a fixed point $A$ on a rough horizontal surface. The other end of the string is attached to a particle $P$ of mass $m$. The particle and string rest on the surface. The coefficient of friction between $P$ and the surface is $\mu$. The particle $P$ is initially held in equilibrium at a distance $\frac{4}{3}a$ from $A$. The particle is then released from rest.

A uniform lamina $OABCD$ is in the form of a rectangle, $OBCD$, joined along the edge $OB$ to a quarter circle $OAB$. The length of $DO$ is $ka$ and the length of $OB$ is $a$. The lamina rests in a vertical plane with its edge $CB$ on a horizontal surface (see diagram).

A particle $Q$ is initially positioned at a distance $d$ vertically above a particle $P$. Particle $P$ is projected with speed $U$ at an angle $\alpha$ above the horizontal. At the same time, $Q$ is projected at an angle $\beta$ below the horizontal. Both particles move freely under gravity. The particles collide at time $T$ after the projections.

Two uniform smooth spheres, $A$ and $B$, of equal radii are on a horizontal surface. They have masses of $m$ and $km$ respectively. Sphere $A$ is moving with speed $u$ at an angle $\alpha$ with the line of centres when it collides with sphere $B$ which is stationary. Immediately after the collision, sphere $A$ moves with speed $v$ at an angle $2\alpha$ with the line of centres (see diagram). It is given that $\tan \alpha =\frac{3}{4}$.

A fixed smooth sphere with radius $a$ and centre $O$ rests on horizontal ground. A particle is projected horizontally from the highest point, $A$, of the sphere with speed $u$. The particle begins to move in a vertical circle along the surface of the sphere. The particle loses contact with the sphere at the point $B$, where the angle $AOB$ is $\theta$. After leaving the surface of the sphere, the particle moves freely under gravity before striking the horizontal ground with speed $3u$ at an angle $\beta$ to the horizontal (see diagram). Find the value of $\beta$.

A particle $P$ of mass $m\text{ kg}$ moving along a rough horizontal table has displacement $x\text{ m}$ from a fixed point $O$ on the table and velocity $v{\text{ ms}}^{-1}$ at time $t\text{ s}$. The particle $P$ is subject to a resistive force of magnitude $mgkv\text{ N}$, where $k$ is a positive constant, and a frictional force of magnitude $\mu mg$. The particle $P$ is initially at $O$ with speed $U{\text{ ms}}^{-1}$.