科目：进阶数学 Further Mathematics (9231)
年份：2020 冬季
试卷：Paper 31
总页数：14
章节：9231 Further Mathematics
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1 A particle $P$ of mass $m$ is placed on a fixed smooth plane which is inclined at an angle $\theta$ to the horizontal. A light spring, of natural length $a$ and modulus of elasticity $3mg$, has one end attached to $P$ and the other end attached to a fixed point $O$ at the top of the plane. The spring lies along a line of greatest slope of the plane. The system is released from rest with the spring at its natural length. Find, in terms of $a$ and $\theta$, an expression for the greatest extension of the spring in the subsequent motion. [3]
A particle $P$ 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 with the string taut and making an angle $\theta$ with the downward vertical. The particle $P$ is then projected with speed $\frac{4}{5}\sqrt{5ag}$ perpendicular to the string and just completes a vertical circle (see diagram).
Find the value of $\cos \theta$. [5]3 One end of a light elastic string, of natural length $a$ and modulus of elasticity $4mg$, is attached to a fixed point $O$. The other end of the string is attached to a particle of mass $m$. The particle moves in a horizontal circle with a constant angular speed $\sqrt{\frac{g}{a}}$ with the string inclined at an angle $\theta$ to the downward vertical through $O$. The length of the string during this motion is $(k+1)a$.
(a) Find the value of $k$.
[4]
(b) Find the value of $\cos \theta$.The diagram shows the cross-section $ABCD$ of a uniform solid object which is formed by removing a cone with cross-section $DCE$ from the top of a larger cone with cross-section $ABE$. The perpendicular distance between $AB$ and $DC$ is $h$, the diameter $AB$ is $6r$ and the diameter $DC$ is $2r$. (a) Find an expression, in terms of $h$, for the distance of the centre of mass of the solid object from $AB$. [4]
The object is freely suspended from the point $B$ and hangs in equilibrium. The angle between $AB$ and the downward vertical through $B$ is $\theta$.
(b) Given that $h = \frac{13}{4} r$, find the value of $\tan \theta$. [2]5 A particle $P$ is projected with speed $u$ at an angle $\alpha$ above the horizontal from a point $O$ on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of $P$ from $O$ at a subsequent time $t$ are denoted by $x$ and $y$ respectively.
(a) Derive the equation of the trajectory of $P$ in the form
$$y = x \tan \alpha - \frac{gx^2}{2u^2} \sec^2 \alpha.$$
[3]
The point $Q$ is the highest point on the trajectory of $P$ in the case where $\alpha = 45^\circ$.
(b) Show that the $x$-coordinate of $Q$ is $\frac{u^2}{2g}$.(c) Find the other value of $\alpha$ for which $P$ would pass through the point $Q$. [4]
6 Two smooth spheres $A$ and $B$ have equal radii and masses $m$ and $2m$ respectively. Sphere $B$ is at rest on a smooth horizontal floor. Sphere $A$ is moving on the floor with velocity $u$ and collides directly with $B$. The coefficient of restitution between the spheres is $e$.
(a) Find, in terms of $u$ and $e$, the velocities of $A$ and $B$ after the collision. [3]
Subsequently, $B$ collides with a fixed vertical wall which makes an angle $\theta$ with the direction of motion of $B$, where $\tan \theta = \frac{3}{4}$.
The coefficient of restitution between $B$ and the wall is $\frac{2}{3}$. Immediately after $B$ collides with the wall, the kinetic energy of $A$ is $\frac{5}{32}$ of the kinetic energy of $B$.
(b) Find the possible values of $e$. [7]11 © UCLES 2020 9231/31/O/N/20 [Turn over]
7 A particle $P$ moving in a straight line has displacement $x$ m from a fixed point $O$ on the line at time $t$ s. The acceleration of $P$, in ms$^{-2}$, is given by $\frac{200}{x^2} - \frac{100}{x^3}$ for $x > 0$. When $t = 0$, $x = 1$ and $P$ has velocity $10$ ms$^{-1}$ directed towards $O$.
(a) Show that the velocity $v$ ms$^{-1}$ of $P$ is given by $v = \frac{10(1-2x)}{x}$. [5](b) Show that $x$ and $t$ are related by the equation $e^{-40t} = (2x - 1)e^{2x-2}$ and deduce what happens to $x$ as $t$ becomes large. [5]Additional Page If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
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