科目：进阶数学 Further Mathematics (9231)
年份：2023 夏季
试卷：Paper 33
总页数：14
章节：9231 Further Mathematics
1. 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 $\alpha$ 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 $\sqrt{3ag}$. It then starts to move along a circular path in a vertical plane. The string goes slack when $P$ is at $B$, where $OB$ makes an angle $\theta$ with the upward vertical.
Given that $\cos \alpha = \frac{4}{5}$, find the value of $\cos \theta$. [4]2 One end of a light elastic string, of natural length $a$ and modulus of elasticity $\lambda mg$, is attached to a fixed point $O$. The string lies on a smooth horizontal surface. A particle $P$ of mass $m$ is attached to the other end of the string. The particle $P$ is projected in the direction $OP$. When the length of the string is $\frac{4}{3}a$, the speed of $P$ is $\sqrt{2ag}$. When the length of the string is $\frac{5}{3}a$, the speed of $P$ is $\frac{1}{2}\sqrt{2ag}$.
Find the value of $\lambda$. [4]A uniform lamina is in the form of a triangle $ABC$, with $AC = 8a$, $BC = 6a$ and angle $ACB = 90^\circ$. The point $D$ on $AC$ is such that $AD = 3a$. The point $E$ on $CB$ is such that $CE = x$ (see diagram). The triangle $CDE$ is removed from the lamina. (a) Find, in terms of $a$ and $x$, the distance of the centre of mass of the remaining object $ADEB$ from $AC$. [4]
The object $ADEB$ is on the point of toppling about the point $E$ when the object is in the vertical plane with its edge $EB$ on a smooth horizontal surface. (b) Find $x$ in terms of $a$. [3]
Two identical smooth uniform spheres $A$ and $B$ each have mass $m$. The two spheres are moving on a smooth horizontal surface when they collide with speeds $u$ and $2u$ respectively. Immediately before the collision, $A$'s direction of motion makes an angle of 30° with the line of centres, and $B$'s direction of motion is perpendicular to the line of centres (see diagram). After the collision, $A$ and $B$ are moving in the same direction. The coefficient of restitution between the spheres is $e$. (a) Find the value of $e$. [5]
(b) Find the loss in the total kinetic energy of the spheres as a result of the collision. [3]
5 One end of a light elastic string, of natural length $12a$ and modulus of elasticity $kmg$, is attached to a fixed point $O$. The other end of the string is attached to a particle of mass $m$. The particle moves with constant speed $\frac{3}{2}\sqrt{3ag}$ in a horizontal circle with centre at a distance $12a$ below $O$. The string is inclined at an angle $\theta$ to the downward vertical through $O$.
(a) Find, in terms of $a$, the extension of the string. [5](b) Find the value of $k$. [3]
6 A particle of mass $mkg$ falls vertically under gravity, from rest. At time $ts$, $P$ has fallen $xm$ and has velocity $vm s^{-1}$. The only forces acting on $P$ are its weight and a resistance of magnitude $kmgv$ N, where $k$ is a constant.
(a) Find an expression for $v$ in terms of $t, g$ and $k$. [5](b) Given that $k = 0.05$, find, in metres, how far $P$ has fallen when its speed is $12 \text{ ms}^{-1}$. [5]7 The points $O$ and $P$ are on a horizontal plane, a distance 8 m apart. A ball is thrown from $O$ with speed $u \text{ ms}^{-1}$ at an angle $\theta$ above the horizontal, where $\tan \theta = \frac{4}{3}$. At the same instant, a model aircraft is launched with speed $5 \text{ ms}^{-1}$ parallel to the horizontal plane from a point 4 m vertically above $P$. The model aircraft moves in the same vertical plane as the ball and in the same horizontal direction as the ball. The model aircraft moves horizontally with a constant speed of $5 \text{ ms}^{-1}$. After $T$ s, the ball and the model aircraft collide.
(a) Find the value of $T$. [6](b) Find the direction in which the ball is moving immediately before the collision. [3]
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