Part IA, 2007, Paper 4

# Part IA, 2007, Paper 4

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Paper 4, Section I, C

commentSketch the graph of $y=3 x^{2}-2 x^{3}$.

A particle of unit mass moves along the $x$ axis in the potential $V(x)=3 x^{2}-2 x^{3}$. Sketch the phase plane, and describe briefly the motion of the particle on the different trajectories.

Paper 4, Section I, C

commentA rocket, moving vertically upwards, ejects gas vertically downwards at speed $u$ relative to the rocket. Derive, giving careful explanations, the equation of motion

$m \frac{d v}{d t}=-u \frac{d m}{d t}-g m$

where $v$ and $m$ are the speed and total mass of the rocket (including fuel) at time $t$.

If $u$ is constant and the rocket starts from rest with total mass $m_{0}$, show that

$m=m_{0} e^{-(g t+v) / u}$

Paper 4, Section II, $12 \mathrm{C}$

commentThe $i$ th particle of a system of $N$ particles has mass $m_{i}$ and, at time $t$, position vector $\mathbf{r}_{i}$ with respect to an origin $O$. It experiences an external force $\mathbf{F}_{i}^{e}$, and also an internal force $\mathbf{F}_{i j}$ due to the $j$ th particle (for each $j=1, \ldots, N, j \neq i$ ), where $\mathbf{F}_{i j}$ is parallel to $\mathbf{r}_{i}-\mathbf{r}_{j}$ and Newton's third law applies.

(i) Show that the position of the centre of mass, $\mathbf{X}$, satisfies

$M \frac{d^{2} \mathbf{X}}{d t^{2}}=\mathbf{F}^{e}$

where $M$ is the total mass of the system and $\mathbf{F}^{e}$ is the sum of the external forces.

(ii) Show that the total angular momentum of the system about the origin, $\mathbf{L}$, satisfies

$\frac{d \mathbf{L}}{d t}=\mathbf{N}$

where $\mathbf{N}$ is the total moment about the origin of the external forces.

(iii) Show that $\mathbf{L}$ can be expressed in the form

$\mathbf{L}=M \mathbf{X} \times \mathbf{V}+\sum_{i} m_{i} \mathbf{r}_{i}^{\prime} \times \mathbf{v}_{i}^{\prime}$

where $\mathbf{V}$ is the velocity of the centre of mass, $\mathbf{r}_{i}^{\prime}$ is the position vector of the $i$ th particle relative to the centre of mass, and $\mathbf{v}_{i}^{\prime}$ is the velocity of the $i$ th particle relative to the centre of mass.

(iv) In the case $N=2$ when the internal forces are derived from a potential $U(|\mathbf{r}|)$, where $\mathbf{r}=\mathbf{r}_{1}-\mathbf{r}_{2}$, and there are no external forces, show that

$\frac{d T}{d t}+\frac{d U}{d t}=0$

where $T$ is the total kinetic energy of the system.

Paper 4, Section II, C

commentA particle moves in the gravitational field of the Sun. The angular momentum per unit mass of the particle is $h$ and the mass of the Sun is $M$. Assuming that the particle moves in a plane, write down the equations of motion in polar coordinates, and derive the equation

$\frac{d^{2} u}{d \theta^{2}}+u=k$

where $u=1 / r$ and $k=G M / h^{2}$.

Write down the equation of the orbit ( $u$ as a function of $\theta$ ), given that the particle moves with the escape velocity and is at the perihelion of its orbit, a distance $r_{0}$ from the Sun, when $\theta=0$. Show that

$\sec ^{4}(\theta / 2) \frac{d \theta}{d t}=\frac{h}{r_{0}^{2}}$

and hence that the particle reaches a distance $2 r_{0}$ from the Sun at time $8 r_{0}^{2} /(3 h)$.

Paper 4, Section II, C

commentA particle of mass $m$ experiences, at the point with position vector $\mathbf{r}$, a force $\mathbf{F}$ given by

$\mathbf{F}=-k \mathbf{r}-e \dot{\mathbf{r}} \times \mathbf{B},$

where $k$ and $e$ are positive constants and $\mathbf{B}$ is a constant, uniform, vector field.

(i) Show that $m \dot{\mathbf{r}} \cdot \dot{\mathbf{r}}+k \mathbf{r} \cdot \mathbf{r}$ is constant. Give a physical interpretation of each term and a physical explanation of the fact that $\mathbf{B}$ does not arise in this expression.

(ii) Show that $m(\dot{\mathbf{r}} \times \mathbf{r}) \cdot \mathbf{B}+\frac{1}{2} e(\mathbf{r} \times \mathbf{B}) \cdot(\mathbf{r} \times \mathbf{B})$ is constant.

(iii) Given that the particle was initially at rest at $\mathbf{r}_{0}$, derive an expression for $\mathbf{r} \cdot \mathbf{B}$ at time $t$.

Paper 4, Section II, C

commentA small ring of mass $m$ is threaded on a smooth rigid wire in the shape of a parabola given by $x^{2}=4 a z$, where $x$ measures horizontal distance and $z$ measures distance vertically upwards. The ring is held at height $z=h$, then released.

(i) Show by dimensional analysis that the period of oscillations, $T$, can be written in the form

$T=(a / g)^{1 / 2} G(h / a)$

for some function $G$.

(ii) Show that $G$ is given by

$G(\beta)=2 \sqrt{2} \int_{-1}^{1}\left(\frac{1+\beta u^{2}}{1-u^{2}}\right)^{\frac{1}{2}} d u$

and find, to first order in $h / a$, the period of small oscillations.

Paper 4 , Section II, E

commentState and prove the Inclusion-Exclusion principle.

The keypad on a cash dispenser is broken. To withdraw money, a customer is required to key in a 4-digit number. However, the key numbered 0 will only function if either the immediately preceding two keypresses were both 1 , or the very first key pressed was 2. Explaining your reasoning clearly, use the Inclusion-Exclusion Principle to find the number of 4-digit codes which can be entered.

Paper 4, Section I, $1 E$

comment(i) Use Euclid's algorithm to find all pairs of integers $x$ and $y$ such that

$7 x+18 y=1$

(ii) Show that, if $n$ is odd, then $n^{3}-n$ is divisible by 24 .

Paper 4, Section I, $2 E$

commentFor integers $k$ and $n$ with $0 \leqslant k \leqslant n$, define $\left(\begin{array}{l}n \\ k\end{array}\right)$. Arguing from your definition, show that

$\left(\begin{array}{c} n-1 \\ k \end{array}\right)+\left(\begin{array}{l} n-1 \\ k-1 \end{array}\right)=\left(\begin{array}{l} n \\ k \end{array}\right)$

for all integers $k$ and $n$ with $1 \leqslant k \leqslant n-1$.

Use induction on $k$ to prove that

$\sum_{j=0}^{k}\left(\begin{array}{c} n+j \\ j \end{array}\right)=\left(\begin{array}{c} n+k+1 \\ k \end{array}\right)$

for all non-negative integers $k$ and $n$.

Paper 4, Section II, $6 \mathrm{E}$

commentStating carefully any results about countability you use, show that for any $d \geqslant 1$ the set $\mathbb{Z}\left[X_{1}, \ldots, X_{d}\right]$ of polynomials with integer coefficients in $d$ variables is countable. By taking $d=1$, deduce that there exist uncountably many transcendental numbers.

Show that there exists a sequence $x_{1}, x_{2}, \ldots$ of real numbers with the property that $f\left(x_{1}, \ldots, x_{d}\right) \neq 0$ for every $d \geqslant 1$ and for every non-zero polynomial $f \in \mathbb{Z}\left[X_{1}, \ldots, X_{d}\right]$.

[You may assume without proof that $\mathbb{R}$ is uncountable.]

Paper 4, Section II, $7 \mathbf{E}$

commentLet $x_{n}(n=1,2, \ldots)$ be real numbers.

What does it mean to say that the sequence $\left(x_{n}\right)_{n=1}^{\infty}$ converges?

What does it mean to say that the series $\sum_{n=1}^{\infty} x_{n}$ converges?

Show that if $\sum_{n=1}^{\infty} x_{n}$ is convergent, then $x_{n} \rightarrow 0$. Show that the converse can be false.

Sequences of positive real numbers $x_{n}, y_{n}(n \geqslant 1)$ are given, such that the inequality

$y_{n+1} \leqslant y_{n}-\frac{1}{2} \min \left(x_{n}, y_{n}\right)$

holds for all $n \geqslant 1$. Show that, if $\sum_{n=1}^{\infty} x_{n}$ diverges, then $y_{n} \rightarrow 0$.

Paper 4, Section II, E

comment(i) Let $p$ be a prime number, and let $x$ and $y$ be integers such that $p$ divides $x y$. Show that at least one of $x$ and $y$ is divisible by $p$. Explain how this enables one to prove the Fundamental Theorem of Arithmetic.

[Standard properties of highest common factors may be assumed without proof.]

(ii) State and prove the Fermat-Euler Theorem.

Let $1 / 359$ have decimal expansion $0 \cdot a_{1} a_{2} \ldots$ with $a_{n} \in\{0,1, \ldots, 9\}$. Use the fact that $60^{2} \equiv 10(\bmod 359)$ to show that, for every $n, a_{n}=a_{n+179}$.