Part IA, 2013, Paper 4

# Part IA, 2013, Paper 4

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Paper 4 , Section II, B

comment(i) An inertial frame $S$ has orthonormal coordinate basis vectors $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$. A second frame $S^{\prime}$ rotates with angular velocity $\boldsymbol{\omega}$ relative to $S$ and has coordinate basis vectors $\mathbf{e}_{1}^{\prime}, \mathbf{e}_{2}^{\prime}, \mathbf{e}_{3}^{\prime}$. The motion of $S^{\prime}$ is characterised by the equations $d \mathbf{e}_{i}^{\prime} / d t=\boldsymbol{\omega} \times \mathbf{e}_{i}^{\prime}$ and at $t=0$ the two coordinate frames coincide.

If a particle $P$ has position vector $\mathbf{r}$ show that $\mathbf{v}=\mathbf{v}^{\prime}+\boldsymbol{\omega} \times \mathbf{r}$ where $\mathbf{v}$ and $\mathbf{v}^{\prime}$ are the velocity vectors of $P$ as seen by observers fixed respectively in $S$ and $S^{\prime}$.

(ii) For the remainder of this question you may assume that $\mathbf{a}=\mathbf{a}^{\prime}+2 \boldsymbol{\omega} \times \mathbf{v}^{\prime}+\boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{r})$ where $\mathbf{a}$ and $\mathbf{a}^{\prime}$ are the acceleration vectors of $P$ as seen by observers fixed respectively in $S$ and $S^{\prime}$, and that $\omega$ is constant.

Consider again the frames $S$ and $S^{\prime}$ in (i). Suppose that $\omega=\omega \mathbf{e}_{3}$ with $\omega$ constant. A particle of mass $m$ moves under a force $\mathbf{F}=-4 m \omega^{2} \mathbf{r}$. When viewed in $S^{\prime}$ its position and velocity at time $t=0$ are $\left(x^{\prime}, y^{\prime}, z^{\prime}\right)=(1,0,0)$ and $\left(\dot{x}^{\prime}, \dot{y}^{\prime}, \dot{z}^{\prime}\right)=(0,0,0)$. Find the motion of the particle in the coordinates of $S^{\prime}$. Show that for an observer fixed in $S^{\prime}$, the particle achieves its maximum speed at time $t=\pi /(4 \omega)$ and determine that speed. [Hint: you may find it useful to consider the combination $\zeta=x^{\prime}+i y^{\prime}$.]

Paper 4, Section $I$, B

commentA frame $S^{\prime}$ moves with constant velocity $v$ along the $x$ axis of an inertial frame $S$ of Minkowski space. A particle $P$ moves with constant velocity $u^{\prime}$ along the $x^{\prime}$ axis of $S^{\prime}$. Find the velocity $u$ of $P$ in $S$.

The rapidity $\varphi$ of any velocity $w$ is defined by $\tanh \varphi=w / c$. Find a relation between the rapidities of $u, u^{\prime}$ and $v$.

Suppose now that $P$ is initially at rest in $S$ and is subsequently given $n$ successive velocity increments of $c / 2$ (each delivered in the instantaneous rest frame of the particle). Show that the resulting velocity of $P$ in $S$ is

$c\left(\frac{e^{2 n \alpha}-1}{e^{2 n \alpha}+1}\right)$

where $\tanh \alpha=1 / 2$.

[You may use without proof the addition formulae $\sinh (a+b)=\sinh a \cosh b+\cosh a \sinh b$ and $\cosh (a+b)=\cosh a \cosh b+\sinh a \sinh b$.]

Paper 4, Section I, B

commentA hot air balloon of mass $M$ is equipped with a bag of sand of mass $m=m(t)$ which decreases in time as the sand is gradually released. In addition to gravity the balloon experiences a constant upwards buoyancy force $T$ and we neglect air resistance effects. Show that if $v(t)$ is the upward speed of the balloon then

$(M+m) \frac{d v}{d t}=T-(M+m) g .$

Initially at $t=0$ the mass of sand is $m(0)=m_{0}$ and the balloon is at rest in equilibrium. Subsequently the sand is released at a constant rate and is depleted in a time $t_{0}$. Show that the speed of the balloon at time $t_{0}$ is

$g t_{0}\left(\left(1+\frac{M}{m_{0}}\right) \ln \left(1+\frac{m_{0}}{M}\right)-1\right)$

[You may use without proof the indefinite integral $\int t /(A-t) d t=-t-A \ln |A-t|+C .$ ]

Paper 4, Section II, B

comment(a) Let $S$ with coordinates $(c t, x, y)$ and $S^{\prime}$ with coordinates $\left(c t^{\prime}, x^{\prime}, y^{\prime}\right)$ be inertial frames in Minkowski space with two spatial dimensions. $S^{\prime}$ moves with velocity $v$ along the $x$-axis of $S$ and they are related by the standard Lorentz transformation:

$\left(\begin{array}{c} c t \\ x \\ y \end{array}\right)=\left(\begin{array}{ccc} \gamma & \gamma v / c & 0 \\ \gamma v / c & \gamma & 0 \\ 0 & 0 & 1 \end{array}\right)\left(\begin{array}{c} c t^{\prime} \\ x^{\prime} \\ y^{\prime} \end{array}\right), \quad \text { where } \gamma=\frac{1}{\sqrt{1-v^{2} / c^{2}}} .$

A photon is emitted at the spacetime origin. In $S^{\prime}$ it has frequency $\nu^{\prime}$ and propagates at angle $\theta^{\prime}$ to the $x^{\prime}$-axis.

Write down the 4 -momentum of the photon in the frame $S^{\prime}$.

Hence or otherwise find the frequency of the photon as seen in $S$. Show that it propagates at angle $\theta$ to the $x$-axis in $S$, where

$\tan \theta=\frac{\tan \theta^{\prime}}{\gamma\left(1+\frac{v}{c} \sec \theta^{\prime}\right)}$

A light source in $S^{\prime}$ emits photons uniformly in all directions in the $x^{\prime} y^{\prime}$-plane. Show that for large $v$, in $S$ half of the light is concentrated into a narrow cone whose semi-angle $\alpha$ is given by $\cos \alpha=v / c$.

(b) The centre-of-mass frame for a system of relativistic particles in Minkowski space is the frame in which the total relativistic 3-momentum is zero.

Two particles $A_{1}$ and $A_{2}$ of rest masses $m_{1}$ and $m_{2}$ move collinearly with uniform velocities $u_{1}$ and $u_{2}$ respectively, along the $x$-axis of a frame $S$. They collide, coalescing to form a single particle $A_{3}$.

Determine the velocity of the centre-of-mass frame of the system comprising $A_{1}$ and $A_{2}$.

Find the speed of $A_{3}$ in $S$ and show that its rest mass $m_{3}$ is given by

$m_{3}^{2}=m_{1}^{2}+m_{2}^{2}+2 m_{1} m_{2} \gamma_{1} \gamma_{2}\left(1-\frac{u_{1} u_{2}}{c^{2}}\right),$

where $\gamma_{i}=\left(1-u_{i}^{2} / c^{2}\right)^{-1 / 2}$

Paper 4, Section II, B

comment(a) A rigid body $Q$ is made up of $N$ particles of masses $m_{i}$ at positions $\mathbf{r}_{i}(t)$. Let $\mathbf{R}(t)$ denote the position of its centre of mass. Show that the total kinetic energy of $Q$ may be decomposed into $T_{1}$, the kinetic energy of the centre of mass, plus a term $T_{2}$ representing the kinetic energy about the centre of mass.

Suppose now that $Q$ is rotating with angular velocity $\boldsymbol{\omega}$ about its centre of mass. Define the moment of inertia $I$ of $Q$ (about the axis defined by $\boldsymbol{\omega}$ ) and derive an expression for $T_{2}$ in terms of $I$ and $\omega=|\omega|$.

(b) Consider a uniform rod $A B$ of length $2 l$ and mass $M$. Two such rods $A B$ and $B C$ are freely hinged together at $B$. The end $A$ is attached to a fixed point $O$ on a perfectly smooth horizontal floor and $A B$ is able to rotate freely about $O$. The rods are initially at rest, lying in a vertical plane with $C$ resting on the floor and each rod making angle $\alpha$ with the horizontal. The rods subsequently move under gravity in their vertical plane.

Find an expression for the angular velocity of rod $A B$ when it makes angle $\theta$ with the floor. Determine the speed at which the hinge strikes the floor.

Paper 4, Section II, B

comment(a) A particle $P$ of unit mass moves in a plane with polar coordinates $(r, \theta)$. You may assume that the radial and angular components of the acceleration are given by $\left(\ddot{r}-r \dot{\theta}^{2}, r \ddot{\theta}+2 \dot{r} \dot{\theta}\right)$, where the dot denotes $d / d t$. The particle experiences a central force corresponding to a potential $V=V(r)$.

(i) Prove that $l=r^{2} \dot{\theta}$ is constant in time and show that the time dependence of the radial coordinate $r(t)$ is equivalent to the motion of a particle in one dimension $x$ in a potential $V_{\text {eff }}$ given by

$V_{\text {eff }}=V(x)+\frac{l^{2}}{2 x^{2}}$

(ii) Now suppose that $V(r)=-e^{-r}$. Show that if $l^{2}<27 / e^{3}$ then two circular orbits are possible with radii $r_{1}<3$ and $r_{2}>3$. Determine whether each orbit is stable or unstable.

(b) Kepler's first and second laws for planetary motion are the following statements:

K1: the planet moves on an ellipse with a focus at the Sun;

K2: the line between the planet and the Sun sweeps out equal areas in equal times.

Show that K2 implies that the force acting on the planet is a central force.

Show that K2 together with $\mathbf{K 1}$ implies that the force is given by the inverse square law.

[You may assume that an ellipse with a focus at the origin has polar equation $\frac{A}{r}=1+\varepsilon \cos \theta$ with $A>0$ and $0<\varepsilon<1$.]

Paper 4, Section I, E

commentLet $\left(x_{n}\right)_{n=1}^{\infty}$ be a sequence of real numbers. What does it mean to say that the sequence $\left(x_{n}\right)$ is convergent? What does it mean to say the series $\sum x_{n}$ is convergent? Show that if $\sum x_{n}$ is convergent, then the sequence $\left(x_{n}\right)$ converges to zero. Show that the converse is not necessarily true.

Paper 4, Section I, E

commentLet $m$ and $n$ be positive integers. State what is meant by the greatest common divisor $\operatorname{gcd}(m, n)$ of $m$ and $n$, and show that there exist integers $a$ and $b$ such that $\operatorname{gcd}(m, n)=a m+b n$. Deduce that an integer $k$ divides both $m$ and $n$ only if $k$ divides $\operatorname{gcd}(m, n)$.

Prove (without using the Fundamental Theorem of Arithmetic) that for any positive integer $k, \operatorname{gcd}(k m, k n)=k \operatorname{gcd}(m, n)$.

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

comment(i) What does it mean to say that a set is countable? Show directly from your definition that any subset of a countable set is countable, and that a countable union of countable sets is countable.

(ii) Let $X$ be either $\mathbb{Z}$ or $\mathbb{Q}$. A function $f: X \rightarrow \mathbb{Z}$ is said to be periodic if there exists a positive integer $n$ such that for every $x \in X, f(x+n)=f(x)$. Show that the set of periodic functions from $\mathbb{Z}$ to itself is countable. Is the set of periodic functions $f: \mathbb{Q} \rightarrow \mathbb{Z}$ countable? Justify your answer.

(iii) Show that $\mathbb{R}^{2}$ is not the union of a countable collection of lines.

[You may assume that $\mathbb{R}$ and the power set of $\mathbb{N}$ are uncountable.]

Paper 4, Section II, E

commentLet $p$ be a prime number, and $x, n$ integers with $n \geqslant 1$.

(i) Prove Fermat's Little Theorem: for any integer $x, x^{p} \equiv x(\bmod p)$.

(ii) Show that if $y$ is an integer such that $x \equiv y\left(\bmod p^{n}\right)$, then for every integer $r \geqslant 0$,

$x^{p^{r}} \equiv y^{p^{r}}\left(\bmod p^{n+r}\right)$

Deduce that $x^{p^{n}} \equiv x^{p^{n-1}}\left(\bmod p^{n}\right) .$

(iii) Show that there exists a unique integer $y \in\left\{0,1, \ldots, p^{n}-1\right\}$ such that

$y \equiv x \quad(\bmod p) \quad \text { and } \quad y^{p} \equiv y \quad\left(\bmod p^{n}\right)$

Paper 4, Section II, E

comment(i) Let $N$ and $r$ be integers with $N \geqslant 0, r \geqslant 1$. Let $S$ be the set of $(r+1)$-tuples $\left(n_{0}, n_{1}, \ldots, n_{r}\right)$ of non-negative integers satisfying the equation $n_{0}+\cdots+n_{r}=N$. By mapping elements of $S$ to suitable subsets of $\{1, \ldots, N+r\}$ of size $r$, or otherwise, show that the number of elements of $S$ equals

$\left(\begin{array}{c} N+r \\ r \end{array}\right)$

(ii) State the Inclusion-Exclusion principle.

(iii) Let $a_{0}, \ldots, a_{r}$ be positive integers. Show that the number of $(r+1)$-tuples $\left(n_{i}\right)$ of integers satisfying

$n_{0}+\cdots+n_{r}=N, \quad 0 \leqslant n_{i}<a_{i} \text { for all } i$

$\begin{aligned} \left(\begin{array}{c} N+r \\ r \end{array}\right) &-\sum_{0 \leqslant i \leqslant r}\left(\begin{array}{c} N+r-a_{i} \\ r \end{array}\right)+\sum_{0 \leqslant i<j \leqslant r}\left(\begin{array}{c} N+r-a_{i}-a_{j} \\ r \end{array}\right) \\ &-\sum_{0 \leqslant i<j<k \leqslant r}\left(\begin{array}{c} N+r-a_{i}-a_{j}-a_{k} \\ r \end{array}\right)+\cdots \end{aligned}$

where the binomial coefficient $\left(\begin{array}{c}m \\ r\end{array}\right)$ is defined to be zero if $m<r$.

Paper 4, Section II, E

comment(i) What does it mean to say that a function $f: X \rightarrow Y$ is injective? What does it mean to say that $f$ is surjective? Let $g: Y \rightarrow Z$ be a function. Show that if $g \circ f$ is injective, then so is $f$, and that if $g \circ f$ is surjective, then so is $g$.

(ii) Let $X_{1}, X_{2}$ be two sets. Their product $X_{1} \times X_{2}$ is the set of ordered pairs $\left(x_{1}, x_{2}\right)$ with $x_{i} \in X_{i}(i=1,2)$. Let $p_{i}$ (for $\left.i=1,2\right)$ be the function

$p_{i}: X_{1} \times X_{2} \rightarrow X_{i}, \quad p_{i}\left(x_{1}, x_{2}\right)=x_{i}$

When is $p_{i}$ surjective? When is $p_{i}$ injective?

(iii) Now let $Y$ be any set, and let $f_{1}: Y \rightarrow X_{1}, f_{2}: Y \rightarrow X_{2}$ be functions. Show that there exists a unique $g: Y \rightarrow X_{1} \times X_{2}$ such that $f_{1}=p_{1} \circ g$ and $f_{2}=p_{2} \circ g$.

Show that if $f_{1}$ or $f_{2}$ is injective, then $g$ is injective. Is the converse true? Justify your answer.

Show that if $g$ is surjective then both $f_{1}$ and $f_{2}$ are surjective. Is the converse true? Justify your answer.