科目：进阶数学 Further Mathematics (9231)
年份：2021 冬季
试卷：Paper 32
总页数：14
章节：9231 Further Mathematics
1 A particle is projected with speed $u$ at an angle $\alpha$ above the horizontal from a point $O$ on a horizontal plane. The particle moves freely under gravity.
(a) Write down the horizontal and vertical components of the velocity of the particle at time $T$ after projection. [2]
At time $T$ after projection, the direction of motion of the particle is perpendicular to the direction of projection.
(b) Express $T$ in terms of $u, g$ and $\alpha$. [2]
(c) Deduce that $T > \frac{u}{g}$. [1]2 A light spring $AB$ has natural length $a$ and modulus of elasticity $5mg$. The end $A$ of the spring is attached to a fixed point on a smooth horizontal surface. A particle $P$ of mass $m$ is attached to the end $B$ of the spring. The spring and particle $P$ are at rest on the surface.
Another particle $Q$ of mass $km$ is moving with speed $\sqrt{4ga}$ along the horizontal surface towards $P$ in the direction $BA$. The particles $P$ and $Q$ collide directly and coalesce. In the subsequent motion the greatest amount by which the spring is compressed is $\frac{1}{3}a$.
Find the value of $k$. [6]Particles $A$ and $B$, of masses $m$ and $3m$ respectively, are connected by a light inextensible string of length $a$ that passes through a fixed smooth ring $R$. Particle $B$ hangs in equilibrium vertically below the ring. Particle $A$ moves in horizontal circles with speed $\nu$. Particles $A$ and $B$ are at the same horizontal level. The angle between $AR$ and $BR$ is $\theta$ (see diagram).
(a) Show that $\cos \theta = \frac{1}{3}$.
[2]
(b) Find an expression for $\nu$ in terms of $a$ and $g$.5 © UCLES 2021 9231/32/O/N/21 [Turn over]
An object is formed by removing a solid cylinder, of height $ka$ and radius $\frac{1}{2}a$, from a uniform solid hemisphere of radius $a$. The axes of symmetry of the hemisphere and the cylinder coincide and one circular face of the cylinder coincides with the plane face of the hemisphere. $AB$ is a diameter of the circular face of the hemisphere (see diagram).
(a) Show that the distance of the centre of mass of the object from $AB$ is $\frac{3a(2-k^2)}{2(8-3k)}$. [4]When the object is freely suspended from the point $A$, the line $AB$ makes an angle $\theta$ with the downward vertical, where $\tan \theta = \frac{7}{18}$.
(b) Find the possible values of $k$. [3]Two uniform smooth spheres $A$ and $B$ of equal radii have masses $m$ and $\frac{3}{2} m$ respectively. The two spheres are each moving with speed $u$ on a horizontal surface when they collide. Immediately before the collision $A$’s direction of motion is along the line of centres, and $B$’s direction of motion makes an angle of $60^\circ$ with the line of centres (see diagram). The coefficient of restitution between the spheres is $\frac{2}{3}$.
(a) Find the angle through which the direction of motion of $B$ is deflected by the collision. [6](b) Find the loss in the total kinetic energy of the system as a result of the collision. [3]
6 A particle $P$ of mass 2 kg moves along a horizontal straight line. The point $O$ is a fixed point on this line. At time $t$ is the velocity of $P$ is $v \text{ ms}^{-1}$ and the displacement of $P$ from $O$ is $x \text{ m}$.
A force of magnitude $\left(8x - \frac{128}{x^3}\right) \text{ N}$ acts on $P$ in the direction $OP$. When $t = 0, x = 8$ and $v = -15$.
(a) Show that $v = -\frac{2}{x}(x^2 - 4)$. [5](b) Find an expression for $x$ in terms of $t$. [4]
7 One end of a light inextensible string of length $a$ is attached to a fixed point $O$. The other end of the string is attached to a particle $P$ of mass $m$. The particle $P$ is held vertically below $O$ with the string taut and then projected horizontally. When the string makes an angle of $60^\circ$ with the upward vertical, $P$ becomes detached from the string. In its subsequent motion, $P$ passes through the point $A$ which is a distance $a$ vertically above $O$. (a) The speed of $P$ when it becomes detached from the string is $V$. Use the equation of the trajectory of a projectile to find $V$ in terms of $a$ and $g$. [4]
(b) Find, in terms of $m$ and $g$, the tension in the string immediately after $P$ is initially projected horizontally. [4]
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