科目：进阶数学 Further Mathematics (9231)
年份：2023 夏季
试卷：Paper 31
总页数：14
章节：9231 Further Mathematics
1. One end of a light elastic string, of natural length $a$ and modulus of elasticity $3mg$, is attached to a fixed point $O$. The other end of the string is attached to a particle $P$ of mass $m$. The string hangs with $P$ vertically below $O$. The particle $P$ is pulled vertically downwards so that the extension of the string is $2a$. The particle $P$ is then released from rest.
(a) Find the speed of $P$ when it is at a distance $\frac{3}{4}a$ below $O$.
[3]
(b) Find the initial acceleration of $P$ when it is released from rest.
[2]A particle $P$ of mass $m$ is moving with speed $u$ on a fixed smooth horizontal surface. It collides at an angle $\alpha$ with a fixed smooth vertical barrier. After the collision, $P$ moves at an angle $\theta$ with the barrier, where $\tan \theta = \frac{1}{2}$ (see diagram). The coefficient of restitution between $P$ and the barrier is $e$. The particle $P$ loses 20% of its kinetic energy as a result of the collision.
Find the value of $e$. [5]3 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 $P$ is held at the point $A$, where $OA$ makes an angle $\theta$ with the downward vertical through $O$, and with the string taut. The particle $P$ is projected perpendicular to $OA$ in an upwards direction with speed $u$. It then starts to move along a circular path in a vertical plane. The string goes slack when $P$ is at $B$, where angle $AOB$ is $90^\circ$ and the speed of $P$ is $\sqrt{\frac{4}{3}ag}$.
(a) Find the value of $\sin \theta$.
[2]
(b) Find, in terms of $m$ and $g$, the tension in the string when $P$ is at $A$. [5]5 © UCLES 2023 9231/31/M/J/23 [Turn over]
An object is formed from a solid hemisphere, of radius $2a$, and a solid cylinder, of radius $a$ and height $d$. The hemisphere and the cylinder are made of the same material. The cylinder is attached to the plane face of the hemisphere. The line $OC$ forms a diameter of the base of the cylinder, where $C$ is the centre of the plane face of the hemisphere and $O$ is common to both circumferences (see diagram). Relative to axes through $O$, parallel and perpendicular to $OC$ as shown, the centre of mass of the object is $(\bar{x}, \bar{y})$.
(a) Show that $\bar{x} = \frac{32a^2 + 3ad}{16a + 3d}$ and find an expression, in terms of $a$ and $d$, for $\bar{y}$. [5]The object is placed on a rough plane which is inclined to the horizontal at an angle $\theta$ where $\sin \theta = \frac{1}{6}$. The object is in equilibrium with $CO$ horizontal, where $CO$ lies in a vertical plane through a line of greatest slope.
(b) Find $d$ in terms of $a$. [3]5 A light elastic string of natural length $a$ and modulus of elasticity $\lambda mg$ has one end attached to a fixed point $O$ on a smooth horizontal surface. When a particle of mass $m$ is attached to the free end of the string, it moves with speed $v$ in a horizontal circle with centre $O$ and radius $x$. When, instead, a particle of mass $2m$ is attached to the free end of the string, this particle moves with speed $\frac{1}{2}v$ in a horizontal circle with centre $O$ and radius $\frac{3}{4}x$.
(a) Find $x$ in terms of $a$. [5](b) Given that $v = \sqrt{12ag}$, find the value of $\lambda$. [2]6 A particle $P$ moving in a straight line has displacement $x\text{ m}$ from a fixed point $O$ on the line and velocity $\nu \text{ m s}^{-1}$ at time $t\text{ s}$. The acceleration of $P$, in $\text{m s}^{-2}$, is given by $6\nu \sqrt{\nu+9}$. When $t=0, x=2$ and $\nu=72$.
(a) Find an expression for $\nu$ in terms of $x$. [4](b) Find an expression for $x$ in terms of $t$. [5]
7 At time $t$ s, a particle $P$ is projected with speed $40 \text{ m/s}^{-1}$ at an angle $\theta$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. The greatest height achieved by $P$ during its flight is $H\text{ m}$ and the corresponding time is $T\text{ s}$.
(a) Obtain expressions for $H$ and $T$ in terms of $\theta$.
During the time between $t = T$ and $t = 3$, $P$ descends a distance $\frac{1}{4} H$.
(b) Find the value of $\theta$.(c) Find the speed of $P$ when $t = 3$. [3]
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