科目：进阶数学 Further Mathematics (9231)
年份：2024 夏季
试卷：Paper 32
总页数：14
章节：9231 Further Mathematics
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 $\tan \theta$. [6]3 © UCLES 2024 9231/32/M/J/24 [Turn over]
2 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 $AB$.
(a) Find $\lambda$ in terms of $m$ and $g$. [3]The particle $P$ is pulled down vertically and released from rest at a distance $\frac{8}{3}a$ below $M$.
(b) Find, in terms of $a$ and $g$, the speed of $P$ as it passes through $M$ in the subsequent motion. [4]3 At time $t = 0$ seconds, a particle $P$ is projected with speed $u \text{ m s}^{-1}$ 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 $u$. [5]7 © UCLES 2024 9231/32/M/J/24 [Turn over]
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 $\mu$.
(a) Find, in terms of $W$, the tension in the string $PQ$. [4](b) Find the value of $\mu$. [3]
5 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) Find the tension in the string, in terms of $m$ and $g$. [4](b) Find the value of $k$. [3]
6 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 $\nu$ m s$^{-1}$ at time $t$ s. The only horizontal force acting on $P$ has magnitude $\frac{1}{10}(2\nu - 1)^2 \mathrm{e}^{-t} \mathrm{N}$ and acts towards $O$. When $t = 0$, $x = 1$ and $\nu = 3$.
(a) Find an expression for $\nu$ in terms of $t$. [5](b) Find an expression for $x$ in terms of $t$. [4]
7 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 $O$.
(a) Show that $\cos \theta = \frac{u^2 + 2ag}{3ag}$. [4]It is given that $\cos \theta = \frac{5}{6}$.
(b) 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]
(c) 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]