科目：进阶数学 Further Mathematics (9231)
年份：2025 冬季
试卷：Paper 32
总页数：14
章节：9231 Further Mathematics
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1. 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 $\omega$. [4]
2 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) Given that the string never becomes slack, find the minimum value of $\mu$. [3]It is now given that $\mu = \frac{1}{2}$.
(b) Find the extension of the string when the particle comes to rest. [3]3 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) Find, in terms of $k, a$ and $\pi$, an expression for the distance of the centre of mass above the horizontal surface. [You may use without proof the result for the centre of mass of a circular sector in the list of formulae (MF19).]
The lamina is on the point of toppling about $B$. (b) Find the value of $k$. [2]
4 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. (a) Show that $d = UT(\sin \alpha + \cos \alpha \tan \beta)$. [4]
The particles collide when $P$ is at its maximum height. (b) Given that $\alpha = 30^\circ$ and $\beta = 60^\circ$, find $d$ in terms of $U$ and $g$. [3]
5 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) Find $v$ in terms of $u$.
[2]
(b) Find the coefficient of restitution between the spheres in terms of $k$. [4](c) Find the range of possible values of $k$. [3]
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 $\beta$. [8]
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7 A particle $P$ of mass $m$ kg moving along a rough horizontal table has displacement $x$ m from a fixed point $O$ on the table and velocity $v$ m/s$^{-1}$ at time $t$ s. The particle $P$ is subject to a resistive force of magnitude $mgkv$ N, where $k$ is a positive constant, and a frictional force of magnitude $\mu mg$. The particle $P$ is initially at $O$ with speed $U$ m/s$^{-1}$.
(a) Show that $t = \frac{1}{gk} \ln \left( \frac{kU + \mu}{kv + \mu} \right)$.
[4]It is given that $U = 10$, $k = 0.04$ and $\mu = 0.2$. (b) Find the distance $P$ moves before coming to rest. [4] (c) Find the average speed of $P$ over the period it is moving. [2]