科目：进阶数学 Further Mathematics (9231)
年份：2012 冬季
试卷：Paper 22
总页数：7
章节：9231 Further Mathematics
A rigid body consists of two uniform circular discs, each of mass $m$ and radius $a$, the centres of which are rigidly attached to the ends $A$ and $B$ of a uniform rod of mass $3m$ and length $10a$. The discs and the rod are in the same plane and $O$ is the point on the rod such that $AO = 4a$ (see diagram). Show that the moment of inertia of the body about an axis through $O$ perpendicular to the plane of the discs is $81ma^2$.
A uniform disc of radius $0.4\text{ m}$ is free to rotate without friction in a vertical plane about a horizontal axis through its centre. The moment of inertia of the disc about the axis is $0.2\text{ kg m}^2$. One end of a light inextensible string is attached to a point on the rim of the disc and the string is wound round the rim. The other end of the string4 A particle $P$ of mass $2m$, moving on a smooth horizontal plane with speed $u$, strikes a fixed smooth vertical barrier. Immediately before the collision the angle between the direction of motion of $P$ and the barrier is $60^\circ$. The coefficient of restitution between $P$ and the barrier is $\frac{1}{3}$. Show that $P$ loses two-thirds of its kinetic energy in the collision.
Subsequently $P$ collides directly with a particle $Q$ of mass $m$ which is moving on the plane with speed $u$ towards $P$. The magnitude of the impulse acting on each particle in the collision is $\frac{2}{3}mu(1 + \sqrt{3})$.
(i) Show that the speed of $P$ after this collision is $\frac{1}{3}u$.
(ii) Find the exact value of the coefficient of restitution between $P$ and $Q$.
5 A particle $P$ of mas8 The yield of a particular crop on a farm is thought to depend principally on the amount of sunshine during the growing season. For a random sample of 8 years, the average yield, $y$ kilograms per square metre, and the average amount of sunshine per day, $x$ hours, are recorded. The results are given in the following table.
$$\begin{bmatrix}
x & 12.2 & 10.4 & 5.2 & 6.3 & 11.8 & 10.0 & 14.2 & 2.3 \\
y & 15 & 9 & 10 & 7 & 8 & 11 & 12 & 6
\end{bmatrix}$$
$$[\Sigma x = 72.4, \Sigma x^2 = 769.9, \Sigma y = 78, \Sigma y^2 = 820, \Sigma xy = 761.3.]$$
(i) Find the equation of the regression line of $y$ on $x$. [4]
(ii) Find the product moment correlation coefficient. [3]
(iii) Test, at the 5% significance level, whether there is positive correlation between the average yield an10 Answer only one of the following two alternatives.
EITHER
Two identical uniform rough spheres $A$ and $B$, each of weight $W$ and radius $a$, are at rest on a rough horizontal plane, and are not in contact with each other. A third identical sphere $C$ rests on $A$ and $B$ with its centre in the same vertical plane as the centres of $A$ and $B$. The line joining the centres of $A$ and $C$ and the line joining the centres of $B$ and $C$ are each inclined at an angle $\theta$ to the vertical (see diagram). The coefficient of friction between each sphere and the plane is $\mu$. The coefficient of friction between $C$ and $A$, and between $C$ and $B$, is $\mu'$. The system remains in equilibrium. Show that
$$\mu \geq \frac{\sin \theta}{3(1 + \cos \theta)} \quad \text{and} \quad \mu'BLANK PAGE
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