科目：进阶数学 Further Mathematics (9231)
年份：2025 夏季
试卷：Paper 33
总页数：14
章节：9231 Further Mathematics
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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 moves in a horizontal circle with constant angular speed $\omega$ and with the string inclined at an angle of $\theta$ to the downward vertical.
Given that $\tan \theta = \frac{4}{3}$, find $\omega$ in terms of $a$ and $g$.
[3]2 A particle $P$ of mass $m$ is attached to one end of a light elastic string of natural length $a$ and modulus of elasticity $mg$. The other end of the string is attached to a fixed point $O$ on a rough plane inclined at an angle of $30^\circ$ to the horizontal. The particle $P$ is held at rest at point $O$ before being released. The frictional force acting on $P$ as it slides down the plane is $\frac{11}{30} mg$.
(a) Find, in terms of $a$, the distance that $P$ moves down the plane before coming to rest. [5](b) It is given that $P$ remains at rest in this new position. Find, in terms of $m$ and $g$, the magnitude of the frictional force in this position. [3]
3 A ball of mass $m \text{ kg}$ is projected vertically upwards with initial speed $U \text{ m s}^{-1}$ and moves under gravity. At time $t \text{ s}$ after projection, the ball has travelled a distance $x \text{ m}$ and its speed is $v \text{ m s}^{-1}$. There is a resistive force of magnitude $m k v^2 \text{ N}$, where $k$ is a positive constant.
(a) Show that the distance travelled by the ball when it is moving upwards is $x = \frac{1}{2k} \ln \left( \frac{g + k U^2}{g + k v^2} \right)$. [4]It is given that $k = 0.025$ and that $U = 20$. (b) Find the time taken for the ball to reach its maximum height. [4]
An object consists of a uniform lamina with a particle attached. The uniform lamina $ABCEFD$ of mass $m$ is formed from a rectangle $ABCD$ and an isosceles triangle $CEF$, where $F$ is the midpoint of $CD$. The rectangle has sides $AB = 2a$ and $AD = a$. The triangle $CEF$ has base $a$ and height $2a$. The particle of mass $km$ is attached to the lamina at $E$. The object rests in a vertical plane with its edge $AD$ on horizontal ground (see diagram). Given that the object is on the point of toppling in its vertical plane about the vertex $D$, find the value of $k$. [4]
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A hollow cylinder of radius $r$ is fixed with its axis horizontal. Points $A, B$ and $O$ are in the same vertical plane perpendicular to the axis of the cylinder, with $A$ and $B$ on the smooth inner surface and $O$ on the axis. $OA$ and $OB$ make angles $90^\circ$ and $\alpha$ respectively with the upward vertical through $O$, with $A$ and $B$ on opposite sides of the vertical. A particle of mass $m$ is projected vertically downwards from point $A$ with speed $\sqrt{\frac{3}{2}} rg$ and moves in a vertical circle inside the cylinder (see diagram). The particle loses contact with the cylinder at point $B$.
(a) Find the value of $\alpha$. [4](b) In the subsequent motion find, in terms of $r$, the greatest height above $O$ reached by the particle. [4]
Two identical uniform smooth spheres $A$ and $B$, each with mass $m$, are moving on a horizontal surface with speeds $2u$ and $u$ respectively when they collide. Immediately before the collision, the spheres are moving parallel to each other in opposite directions such that their directions of motion each make an angle $\theta$ with the line of centres (see diagram). As a result of the collision, $B$ moves in a direction which is perpendicular to its initial direction of motion. The coefficient of restitution between the spheres is $e$. (a) Find an expression for $\tan \theta$ in terms of $e$. [6]
As a result of the collision, $A$ moves in a direction which is perpendicular to the line of centres. (b) Find the value of $\theta$. [2]
7 A particle $P$ is projected from a point $O$ with speed $U$ at an angle $45^\circ$ above the horizontal and moves freely under gravity.
(a) State the vertical and horizontal components of velocity at time $t$. [1]
At time $T$, particle $P$ is moving at an angle of $60^\circ$ below the horizontal.
(b) Show that $T = \frac{U}{2g}(\sqrt{2} + \sqrt{6}).$ [3]At time $T$, the particle strikes a smooth horizontal plane at a point which is a horizontal distance $D$ from $O$ and a vertical distance $H$ below $O$.
(c) Find the ratio $H : D$. [4]
After striking the horizontal plane, $P$ rebounds with speed $w$. The coefficient of restitution between $P$ and the plane is $\frac{2}{3}$.
(d) Find $w$ in terms of $U$. [3]