科目：进阶数学 Further Mathematics (9231)
年份：2021 冬季
试卷：Paper 33
总页数：14
章节：9231 Further Mathematics
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1. One end of a light elastic string, of natural length $a$ and modulus of elasticity $3mg$, is attached to a fixed point $O$ on a smooth horizontal plane. A particle $P$ of mass $m$ is attached to the other end of the string and moves in a horizontal circle with centre $O$. The speed of $P$ is $\sqrt{\frac{4}{3}}ga$.
Find the extension of the string. [4]2 A particle $P$ of mass $m$ kg moves along a horizontal straight line with acceleration $a$ ms$^{-2}$ given by
$$a = \frac{v(1-2t^2)}{t},$$
where $v$ ms$^{-1}$ is the velocity of $P$ at time $t$ s.
(a) Find an expression for $v$ in terms of $t$ and an arbitrary constant. [3]
(b) Given that $a = 5$ when $t = 1$, find an expression, in terms of $m$ and $t$, for the horizontal force acting on $P$ at time $t$. [3]3 A light elastic string has natural length $a$ and modulus of elasticity $12mg$. One end of the string is attached to a fixed point $O$. The other end of the string is attached to a particle of mass $m$. The particle hangs in equilibrium vertically below $O$. The particle is pulled vertically down and released from rest with the extension of the string equal to $e$, where $e > \frac{1}{3}a$. In the subsequent motion the particle has speed $\sqrt{2ga}$ when it has ascended a distance $\frac{1}{3}a$.
Find $e$ in terms of $a$. [6]A uniform lamina $AECF$ is formed by removing two identical triangles $BCE$ and $CDF$ from a square lamina $ABCD$. The square has side $3a$ and $EB = DF = h$ (see diagram). (a) Find the distance of the centre of mass of the lamina $AECF$ from $AD$ and from $AB$, giving your answers in terms of $a$ and $h$. [5]
The lamina $AECF$ is placed vertically on its edge $AE$ on a horizontal plane. (b) Find, in terms of $a$, the set of values of $h$ for which the lamina remains in equilibrium. [3]
5 A particle $P$ is projected from a point $O$ on a horizontal plane and moves freely under gravity. Its initial speed is $u \text{ m s}^{-1}$ and its angle of projection is $\sin^{-1}\left(\frac{4}{3}\right)$ above the horizontal. At time 8 s after projection, $P$ is at the point $A$. At time 32 s after projection, $P$ is at the point $B$. The direction of motion of $P$ at $B$ is perpendicular to its direction of motion at $A$.
Find the value of $u$. [7]9 © UCLES 2021 9231/33/O/N/21 [Turn over]
6 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 $P$ moves in complete vertical circles about $O$ with the string taut. The points $A$ and $B$ are on the path of $P$ with $AB$ a diameter of the circle. $OA$ makes an angle $\theta$ with the downward vertical through $O$ and $OB$ makes an angle $\theta$ with the upward vertical through $O$. The speed of $P$ when it is at $A$ is $\sqrt{5ag}$.
The ratio of the tension in the string when $P$ is at $A$ to the tension in the string when $P$ is at $B$ is 9:5.
(a) Find the value of $\cos \theta$. [6](b) Find, in terms of $a$ and $g$, the greatest speed of $P$ during its motion. [2]
The smooth vertical walls $AB$ and $CB$ are at right angles to each other. A particle $P$ is moving with speed $u$ on a smooth horizontal floor and strikes the wall $CB$ at an angle $\alpha$. It rebounds at an angle $\beta$ to the wall $CB$. The particle then strikes the wall $AB$ and rebounds at an angle $\gamma$ to that wall (see diagram). The coefficient of restitution between each wall and $P$ is $e$. (a) Show that $\tan \beta = e \tan \alpha$. [3] (b) Express $\gamma$ in terms of $\alpha$ and explain what this result means about the final direction of motion of $P$.
As a result of the two impacts the particle loses $\frac{8}{9}$ of its initial kinetic energy.
(c) Given that $\alpha + \beta = 90^\circ$, find the value of $e$ and the value of $\tan \alpha$. [4]Additional Page If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
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