科目：进阶数学 Further Mathematics (9231)
年份：2022 冬季
试卷：Paper 31
总页数：14
章节：9231 Further Mathematics
1. A particle of mass 2kg is attached to one end of a light inextensible string of length 0.6m. The other end of the string is attached to a fixed point on a smooth horizontal surface. The particle is moving in a circular path on the surface. The tension in the string is 20N. Find how many revolutions the particle makes per minute. [3]
2 A light elastic string has natural length $a$ and modulus of elasticity $4mg$. One end of the string is fixed to a point $O$ 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 along the surface in the direction $OP$. When the length of the string is $\frac{5}{4}a$, the speed of $P$ is $v$. When the length of the string is $\frac{3}{2}a$, the speed of $P$ is $\frac{1}{2}v$.
(a) Find an expression for $v$ in terms of $a$ and $g$.
(b) Find, in terms of $g$, the acceleration of $P$ when the stretched length of the string is $\frac{3}{2}a$.A smooth cylinder is fixed to a rough horizontal surface with its axis of symmetry horizontal. A uniform rod $AB$, of length $4a$ and weight $W$, rests against the surface of the cylinder. The end $A$ of the rod is in contact with the horizontal surface. The vertical plane containing the rod $AB$ is perpendicular to the axis of the cylinder. The point of contact between the rod and the cylinder is $C$, where $AC = 3a$. The angle between the rod and the horizontal surface is $\theta$ where $\tan \theta = \frac{3}{4}$ (see diagram). The coefficient of friction between the rod and the horizontal surface is $\frac{6}{7}$.
A particle of weight $kW$ is attached to the rod at $B$. The rod is about to slip. The normal reaction between the rod and the cylinder is $N$.
(a) Show that $N = \frac(b) Find the value of $k$. [5]
4 A particle of mass 0.5 kg moves along a horizontal straight line. Its velocity is $v \text{ m s}^{-1}$ at time $t$ s. The forces acting on the particle are a driving force of magnitude 50 N and a resistance of magnitude $2v^2$ N. The initial velocity of the particle is $3 \text{ m s}^{-1}$.
(a) Find an expression for $v$ in terms of $t$. [7](b) Deduce the limiting value of $v$. [1]
5 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 string is held taut with $OP$ horizontal. The particle $P$ is projected vertically downwards with speed $\sqrt{\frac{1}{3}}ag$ and starts to move in a vertical circle. $P$ passes through the lowest point of the circle and reaches the point $Q$ where $OQ$ makes an angle $\theta$ with the downward vertical. At $Q$ the speed of $P$ is $\sqrt{kag}$ and the tension in the string is $\frac{11}{6}mg$.
(a) Find the value of $k$ and the value of $\cos \theta$. [4]At $Q$ the particle $P$ becomes detached from the string. (b) In the subsequent motion, find the greatest height reached by $P$ above the level of the lowest point of the circle. [4]
Two uniform smooth spheres $A$ and $B$ of equal radii have masses $m$ and $km$ respectively. The two spheres are moving on a horizontal surface with speeds $u$ and $\frac{5}{8}u$ respectively. Immediately before the spheres collide, $A$ is travelling along the line of centres, and $B$’s direction of motion makes an angle $\alpha$ with the line of centres (see diagram). The coefficient of restitution between the spheres is $\frac{2}{3}$ and $\tan \alpha = \frac{3}{4}$.
After the collision, the direction of motion of $B$ is perpendicular to the line of centres.
(a) Find the value of $k$. [4](b) Find the loss in the total kinetic energy as a result of the collision. [4]
7 A particle $P$ is projected with speed $V \text{ m/s}^{-1}$ at an angle $75^\circ$ above the horizontal from a point $O$ on a horizontal plane. It then moves freely under gravity.
(a) Show that the total time of flight, in seconds, is $\frac{2V}{g} \sin 75^\circ$. [2]
A smooth vertical barrier is now inserted with its lower end on the plane at a distance 15 m from $O$. The particle is projected as before but now strikes the barrier, rebounds and returns to $O$. The coefficient of restitution between the barrier and the particle is $\frac{3}{5}$.
(b) Explain why the total time of flight is unchanged. [1](c) Find an expression for $V$ in terms of $g$. [7]
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