科目：进阶数学 Further Mathematics (9231)
年份：2026 夏季
试卷：Paper 33
总页数：14
章节：9231 Further Mathematics
1 A particle $P$ of mass $m$ is moving in a horizontal circle with constant angular speed $\omega$ on the smooth inner surface of a hemispherical shell of radius $a$. The plane of the circular motion is at a height $h = \frac{1}{3}a$ above the lowest point on the shell. The angle between the vertical and the normal reaction of the surface on $P$ is $\theta$.
Find $\omega$ in terms of $a$ and $g$. [5]2 A particle $P$ of mass $m$ is moving in a straight horizontal line towards a fixed point $A$ on the line. The particle is subject to a resistive force that is inversely proportional to the square of the distance between $P$ and $A$. Initially, the distance between $P$ and $A$ is $d$ and the speed of $P$ is $u$. When the distance between $P$ and $A$ is $\frac{1}{2} d$, the particle has lost $\frac{3}{4}$ of its kinetic energy.
Find, in terms of $d$, the minimum distance of $P$ from $A$. [6]A fixed smooth solid sphere has centre $O$ and radius $a$. A particle of mass $m$ is projected perpendicular to $OA$ in a downward direction with speed $u$ from the point $A$ on the outer surface of the sphere, where $OA$ makes an angle $\alpha$ with the upward vertical through $O$ (see diagram). The particle moves in part of a vertical circle on the surface of the sphere. It loses contact with the sphere at the point $B$, where $OB$ makes an angle $\beta$ with the upward vertical through $O$. The speed of the particle at the point $B$ is $v$. Given that the ratio of $u$ to $v$ is 2:3 find the ratio $\cos \alpha : \cos \beta$. [5]
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An object consists of a uniform solid cube of side length $d$, and two uniform solid square-based pyramids. The cube and the pyramids are made of the same material. One pyramid, of base length $d$ and height $h$, is fixed to the top face of the cube so that the base of this pyramid and the face of the cube coincide. Point $A$ is at the top vertex of this pyramid. Point $B$ is in the centre of the bottom face of the cube, vertically below $A$. The second pyramid, of base length $d$ and height $kh$, is fixed to a vertical face of the cube so that the base of this pyramid and the vertical face of the cube coincide (see diagram, which shows a cross-section, including the axes of symmetry). (a) Find an expression for the distance of the centre of mass of the object from $AB$ in terms of $
The object is placed on a horizontal plane so that the bottom face of the cube that contains point $B$ is in contact with the plane. It is given that the object is on the point of toppling. (b) Find the value of $h$ in terms of $k$ and $d$. [3] (c) Given that $h = 1.5d$, find the value of $k$. [2]
Two uniform smooth spheres $A$ and $B$ of equal radii and equal masses $m$ are moving with speeds $u_A$ and $u_B$ respectively on a smooth horizontal surface when they collide. Immediately before the collision, the direction of motion of $A$ makes an angle $\alpha$ with the line of centres, and the direction of motion of $B$ makes an angle $\beta$ with the line of centres (see diagram). Immediately after the collision, the direction of motion of $A$ makes an angle $\theta$ with the line of centres, and the direction of motion of $B$ makes an angle $\varphi$ with the line of centres. It is given that $\cos \alpha = \cos \varphi = \frac{4}{5}$ and $\cos \beta = \frac{24}{25}$ and $\cos \theta = \frac{3}{5}$.
(a) Find the percentage loss in the kinetic energy of $A$. [2](b) Find the value of the coefficient of restitution between the spheres. [4]
6 A smooth plane is inclined at an angle $\theta$ to the horizontal, where $\sin \theta = \frac{4}{5}$. One end of a light elastic spring, with natural length $10a$ and modulus of elasticity $1.6mg$, is attached to a fixed point $O$ on the plane. The other end of the spring is attached to a particle $P$ of mass $0.2m$. The point $O$ and the particle $P$ lie on the same line of greatest slope of the plane, with $P$ above $O$. The particle $P$ is at rest.
(a) Show that the distance $OP$ is $9a$.
[3]
A particle $Q$ of mass $0.3m$ is held on the plane on the line of greatest slope through $O$, at a distance $10a$ above $P$. The particle $Q$ is released from rest. The particle $Q$ then collides and coalesces with particle $P$.
(b) Find, in terms of $a$ and $g$, the speed of the comb(c) As a result of the collision, the spring is further compressed. Find, in terms of $a$, the minimum distance of the combined particle from $O$. [4]
7 At time $t = 0$ s, a particle $P$ is projected from a point $O$ on horizontal ground. The initial velocity of $P$ is $25 \text{ m/s}^{-1}$ at an angle $\alpha$ above the horizontal, where $\sin \alpha = \frac{3}{4}$. At time $t = T$ s, a second particle $Q$ is projected vertically upwards from the horizontal ground directly below the point where $P$ will reach its greatest height. The initial velocity of $Q$ is also $25 \text{ m/s}^{-1}$. Both particles move freely under gravity. The two particles collide when $Q$ is travelling upwards and at the point where $P$ reaches its greatest height.
(a) Show that the two particles collide 2 seconds after $P$ is projected. [1]
(b) Find the value of $T$. [3](c) As a result of the collision, the trajectory of $P$ changes, but the horizontal component of its velocity remains unchanged. Immediately after the collision, the speed of $P$ is $\nu m s^{-1}$, at an angle $\theta$ above the horizontal.
Find, in terms of $\tan \theta$, the greatest height of $P$ above the ground after the collision. [4]Additional page If you use the following page to complete the answer to any question, the question number must be clearly shown.
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