Part IA, 2012, Paper 4

# Part IA, 2012, Paper 4

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Paper 4, Section I, B

commentLet $S$ and $S^{\prime}$ be inertial frames in 2-dimensional spacetime with coordinate systems $(t, x)$ and $\left(t^{\prime}, x^{\prime}\right)$ respectively. Suppose that $S^{\prime}$ moves with positive velocity $v$ relative to $S$ and the spacetime origins of $S$ and $S^{\prime}$ coincide. Write down the Lorentz transformation relating the coordinates of any event relative to the two frames.

Show that events which occur simultaneously in $S$ are not generally seen to be simultaneous when viewed in $S^{\prime}$.

In $S$ two light sources $A$ and $B$ are at rest and placed a distance $d$ apart. They simultaneously each emit a photon in the positive $x$ direction. Show that in $S^{\prime}$ the photons are separated by a constant distance $d \sqrt{\frac{c+v}{c-v}}$.

Paper 4, Section I, B

commentTwo particles of masses $m_{1}$ and $m_{2}$ have position vectors $\mathbf{r}_{1}$ and $\mathbf{r}_{2}$ respectively. Particle 2 exerts a force $\mathbf{F}_{12}\left(\mathbf{r}\right.$ ) on particle 1 (where $\mathbf{r}=\mathbf{r}_{1}-\mathbf{r}_{2}$ ) and there are no external forces.

Prove that the centre of mass of the two-particle system will move at constant speed along a straight line.

Explain how the two-body problem of determining the motion of the system may be reduced to that of a single particle moving under the force $\mathbf{F}_{12}$.

Suppose now that $m_{1}=m_{2}=m$ and that

$\mathbf{F}_{12}=-\frac{G m^{2}}{r^{3}} \mathbf{r}$

is gravitational attraction. Let $C$ be a circle fixed in space. Is it possible (by suitable choice of initial conditions) for the two particles to be traversing $C$ at the same constant angular speed? Give a brief reason for your answer.

Paper 4, Section II, B

comment(a) Define the 4-momentum $\mathbf{P}$ of a particle of rest mass $m$ and 3 -velocity $\mathbf{v}$, and the 4-momentum of a photon of frequency $\nu$ (having zero rest mass) moving in the direction of the unit vector $e$.

Show that if $\mathbf{P}_{1}$ and $\mathbf{P}_{2}$ are timelike future-pointing 4-vectors then $\mathbf{P}_{1} \cdot \mathbf{P}_{2} \geqslant 0$ (where the dot denotes the Lorentz-invariant scalar product). Hence or otherwise show that the law of conservation of 4 -momentum forbids a photon to spontaneously decay into an electron-positron pair. [Electrons and positrons have equal rest masses $m>0$.]

(b) In the laboratory frame an electron travelling with velocity u collides with a positron at rest. They annihilate, producing two photons of frequencies $\nu_{1}$ and $\nu_{2}$ that move off at angles $\theta_{1}$ and $\theta_{2}$ to $\mathbf{u}$, in the directions of the unit vectors $\mathbf{e}_{1}$ and $\mathbf{e}_{2}$ respectively. By considering 4-momenta in the laboratory frame, or otherwise, show that

$\frac{1+\cos \left(\theta_{1}+\theta_{2}\right)}{\cos \theta_{1}+\cos \theta_{2}}=\sqrt{\frac{\gamma-1}{\gamma+1}}$

where $\gamma=\left(1-\frac{u^{2}}{c^{2}}\right)^{-1 / 2}$

Paper 4, Section II, B

comment(a) State the parallel axis theorem for moments of inertia.

(b) A uniform circular disc $D$ of radius $a$ and total mass $m$ can turn frictionlessly about a fixed horizontal axis that passes through a point $A$ on its circumference and is perpendicular to its plane. Initially the disc hangs at rest (in constant gravity $g$ ) with its centre $O$ being vertically below $A$. Suppose the disc is disturbed and executes free oscillations. Show that the period of small oscillations is $2 \pi \sqrt{\frac{3 a}{2 g}}$.

(c) Suppose now that the disc is released from rest when the radius $O A$ is vertical with $O$ directly above $A$. Find the angular velocity and angular acceleration of $O$ about $A$ when the disc has turned through angle $\theta$. Let $\mathbf{R}$ denote the reaction force at $A$ on the disc. Find the acceleration of the centre of mass of the disc. Hence, or otherwise, show that the component of $\mathbf{R}$ parallel to $O A$ is $m g(7 \cos \theta-4) / 3$.

Paper 4, Section II, B

commentFor any frame $S$ and vector $\mathbf{A}$, let $\left[\frac{d \mathbf{A}}{d t}\right]_{S}$ denote the derivative of $\mathbf{A}$ relative to $S$. A frame of reference $S^{\prime}$ rotates with constant angular velocity $\omega$ with respect to an inertial frame $S$ and the two frames have a common origin $O$. [You may assume that for any vector $\left.\mathbf{A},\left[\frac{d \mathbf{A}}{d t}\right]_{S}=\left[\frac{d \mathbf{A}}{d t}\right]_{S^{\prime}}+\boldsymbol{\omega} \times \mathbf{A} .\right]$

(a) If $\mathbf{r}(t)$ is the position vector of a point $P$ from $O$, show that

$\left[\frac{d^{2} \mathbf{r}}{d t^{2}}\right]_{S}=\left[\frac{d^{2} \mathbf{r}}{d t^{2}}\right]_{S^{\prime}}+2 \boldsymbol{\omega} \times \mathbf{v}^{\prime}+\boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{r})$

where $\mathbf{v}^{\prime}=\left[\frac{d \mathbf{r}}{d t}\right]_{S^{\prime}}$ is the velocity in $S^{\prime}$.

Suppose now that $\mathbf{r}(t)$ is the position vector of a particle of mass $m$ moving under a conservative force $\mathbf{F}=-\nabla \phi$ and a force $\mathbf{G}$ that is always orthogonal to the velocity $\mathbf{v}^{\prime}$ in $S^{\prime}$. Show that the quantity

$E=\frac{1}{2} m \mathbf{v}^{\prime} \cdot \mathbf{v}^{\prime}+\phi-\frac{m}{2}(\boldsymbol{\omega} \times \mathbf{r}) \cdot(\boldsymbol{\omega} \times \mathbf{r})$

is a constant of the motion. [You may assume that $\nabla[(\boldsymbol{\omega} \times \mathbf{r}) \cdot(\boldsymbol{\omega} \times \mathbf{r})]=-2 \boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{r})$.]

(b) A bead slides on a frictionless circular hoop of radius $a$ which is forced to rotate with constant angular speed $\omega$ about a vertical diameter. Let $\theta(t)$ denote the angle between the line from the centre of the hoop to the bead and the downward vertical. Using the results of (a), or otherwise, show that

$\ddot{\theta}+\left(\frac{g}{a}-\omega^{2} \cos \theta\right) \sin \theta=0 .$

Deduce that if $\omega^{2}>g / a$ there are two equilibrium positions $\theta=\theta_{0}$ off the axis of rotation, and show that these are stable equilibria.

Paper 4, Section II, B

commentLet $(r, \theta)$ be polar coordinates in the plane. A particle of mass $m$ moves in the plane under an attractive force of $m f(r)$ towards the origin $O$. You may assume that the acceleration a is given by

$\mathbf{a}=\left(\ddot{r}-r \dot{\theta}^{2}\right) \hat{\mathbf{r}}+\frac{1}{r} \frac{d}{d t}\left(r^{2} \dot{\theta}\right) \hat{\theta}$

where $\hat{\mathbf{r}}$ and $\hat{\theta}$ are the unit vectors in the directions of increasing $r$ and $\theta$ respectively, and the dot denotes $d / d t$.

(a) Show that $l=r^{2} \dot{\theta}$ is a constant of the motion. Introducing $u=1 / r$ show that $\dot{r}=-l \frac{d u}{d \theta}$ and derive the geometric orbit equation

$l^{2} u^{2}\left(\frac{d^{2} u}{d \theta^{2}}+u\right)=f\left(\frac{1}{u}\right)$

(b) Suppose now that

$f(r)=\frac{3 r+9}{r^{3}}$

and that initially the particle is at distance $r_{0}=1$ from $O$, moving with speed $v_{0}=4$ in a direction making angle $\pi / 3$ with the radial vector pointing towards $O$.

Show that $l=2 \sqrt{3}$ and find $u$ as a function of $\theta$. Hence or otherwise show that the particle returns to its original position after one revolution about $O$ and then flies off to infinity.

Paper 4, Section I, $2 \mathrm{D}$

commentWhat is an equivalence relation on a set $X$ ? If $R$ is an equivalence relation on $X$, what is an equivalence class of $R$ ? Prove that the equivalence classes of $R$ form a partition of $X$.

Let $R$ and $S$ be equivalence relations on a set $X$. Which of the following are always equivalence relations? Give proofs or counterexamples as appropriate.

(i) The relation $V$ on $X$ given by $x V y$ if both $x R y$ and $x S y$.

(ii) The relation $W$ on $X$ given by $x W y$ if $x R y$ or $x S y$.

Paper 4, Section I, D

comment(i) Find integers $x$ and $y$ such that $18 x+23 y=101$.

(ii) Find an integer $x$ such that $x \equiv 3(\bmod 18)$ and $x \equiv 2(\bmod 23)$.

Paper 4, Section II, D

commentShow that there is no injection from the power-set of $\mathbb{R}$ to $\mathbb{R}$. Show also that there is an injection from $\mathbb{R}^{2}$ to $\mathbb{R}$.

Let $X$ be the set of all functions $f$ from $\mathbb{R}$ to $\mathbb{R}$ such that $f(x)=x$ for all but finitely many $x$. Determine whether or not there exists an injection from $X$ to $\mathbb{R}$.

Paper 4, Section II, D

commentProve that each of the following numbers is irrational: (i) $\sqrt{2}+\sqrt{3}$ (ii) $e$ (iii) The real root of the equation $x^{3}+4 x-7=0$ (iv) $\log _{2} 3$.

Paper 4, Section II, D

commentState Fermat's Theorem and Wilson's Theorem.

For which prime numbers $p$ does the equation $x^{2} \equiv-1(\bmod p)$ have a solution? Justify your answer.

For a prime number $p$, and an integer $x$ that is not a multiple of $p$, the order of $x$ $(\bmod p)$ is the least positive integer $d$ such that $x^{d} \equiv 1(\bmod p)$. Show that if $x$ has order $d$ and also $x^{k} \equiv 1(\bmod p)$ then $d$ must divide $k$.

For a positive integer $n$, let $F_{n}=2^{2^{n}}+1$. If $p$ is a prime factor of $F_{n}$, determine the order of $2(\bmod p)$. Hence show that the $F_{n}$ are pairwise coprime.

Show that if $p$ is a prime of the form $4 k+3$ then $p$ cannot be a factor of any $F_{n}$. Give, with justification, a prime $p$ of the form $4 k+1$ such that $p$ is not a factor of any $F_{n}$.

Paper 4, Section II, D

commentLet $X$ be a set, and let $f$ and $g$ be functions from $X$ to $X$. Which of the following are always true and which can be false? Give proofs or counterexamples as appropriate.

(i) If $f g$ is the identity map then $g f$ is the identity map.

(ii) If $f g=g$ then $f$ is the identity map.

(iii) If $f g=f$ then $g$ is the identity map.

How (if at all) do your answers change if we are given that $X$ is finite?

Determine which sets $X$ have the following property: if $f$ is a function from $X$ to $X$ such that for every $x \in X$ there exists a positive integer $n$ with $f^{n}(x)=x$, then there exists a positive integer $n$ such that $f^{n}$ is the identity map. [Here $f^{n}$ denotes the $n$-fold composition of $f$ with itself.]