• # Paper 4, Section I, C

Write down the 4-momentum of a particle with energy $E$ and 3-momentum p. State the relationship between the energy $E$ and wavelength $\lambda$ of a photon.

An electron of mass $m$ is at rest at the origin of the laboratory frame: write down its 4 -momentum. The electron is scattered by a photon of wavelength $\lambda_{1}$ travelling along the $x$-axis: write down the initial 4-momentum of the photon. Afterwards, the photon has wavelength $\lambda_{2}$ and has been deflected through an angle $\theta$. Show that

$\lambda_{2}-\lambda_{1}=\frac{2 h}{m c} \sin ^{2}\left(\frac{1}{2} \theta\right)$

where $c$ is the speed of light and $h$ is Planck's constant.

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• # Paper 4, Section I, C

Find the moment of inertia of a uniform sphere of mass $M$ and radius $a$ about an axis through its centre.

The kinetic energy $T$ of any rigid body with total mass $M$, centre of mass $\mathbf{R}$, moment of inertia $I$ about an axis of rotation through $\mathbf{R}$, and angular velocity $\omega$ about that same axis, is given by $T=\frac{1}{2} M \dot{\mathbf{R}}^{2}+\frac{1}{2} I \omega^{2}$. What physical interpretation can be given to the two parts of this expression?

A spherical marble of uniform density and mass $M$ rolls without slipping at speed $V$ along a flat surface. Explaining any relationship that you use between its speed and angular velocity, show that the kinetic energy of the marble is $\frac{7}{10} M V^{2}$.

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• # Paper 4, Section II, $\mathbf{9 C}$

A particle is projected vertically upwards at speed $V$ from the surface of the Earth, which may be treated as a perfect sphere. The variation of gravity with height should not be ignored, but the rotation of the Earth should be. Show that the height $z(t)$ of the particle obeys

$\ddot{z}=-\frac{g R^{2}}{(R+z)^{2}},$

where $R$ is the radius of the Earth and $g$ is the acceleration due to gravity measured at the Earth's surface.

Using dimensional analysis, show that the maximum height $H$ of the particle and the time $T$ taken to reach that height are given by

$H=R F(\lambda) \quad \text { and } \quad T=\frac{V}{g} G(\lambda)$

where $F$ and $G$ are functions of $\lambda=V^{2} / g R$.

Write down the equation of conservation of energy and deduce that

$T=\int_{0}^{H} \sqrt{\frac{R+z}{V^{2} R-\left(2 g R-V^{2}\right) z}} d z$

Hence or otherwise show that

$F(\lambda)=\frac{\lambda}{2-\lambda} \quad \text { and } \quad G(\lambda)=\int_{0}^{1} \sqrt{\frac{2-\lambda+\lambda x}{(2-\lambda)^{3}(1-x)}} d x$

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• # Paper 4, Section II, C

Write down the Lorentz transform relating the components of a 4-vector between two inertial frames.

A particle moves along the $x$-axis of an inertial frame. Its position at time $t$ is $x(t)$, its velocity is $u=d x / d t$, and its 4 -position is $X=(c t, x)$, where $c$ is the speed of light. The particle's 4-velocity is given by $U=d X / d \tau$ and its 4 -acceleration is $A=d U / d \tau$, where proper time $\tau$ is defined by $c^{2} d \tau^{2}=c^{2} d t^{2}-d x^{2}$. Show that

$U=\gamma(c, u) \quad \text { and } \quad A=\gamma^{4} \dot{u}(u / c, 1)$

where $\gamma=\left(1-u^{2} / c^{2}\right)^{-\frac{1}{2}}$ and $\dot{u}=d u / d t$.

The proper 3-acceleration a of the particle is defined to be the spatial component of its 4-acceleration measured in the particle's instantaneous rest frame. By transforming $A$ to the rest frame, or otherwise, show that

$a=\gamma^{3} \dot{u}=\frac{d}{d t}(\gamma u)$

Given that the particle moves with constant proper 3 -acceleration starting from rest at the origin, show that

$x(t)=\frac{c^{2}}{a}\left(\sqrt{1+\frac{a^{2} t^{2}}{c^{2}}}-1\right)$

and that, if $a t \ll c$, then $x \approx \frac{1}{2} a t^{2}$.

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• # Paper 4, Section II, C

Consider a particle with position vector $r(t)$ moving in a plane described by polar coordinates $(r, \theta)$. Obtain expressions for the radial $(r)$ and transverse $(\theta)$ components of the velocity $\dot{\mathbf{r}}$ and acceleration $\ddot{\mathbf{r}}$.

A charged particle of unit mass moves in the electric field of another charge that is fixed at the origin. The electrostatic force on the particle is $-p / r^{2}$ in the radial direction, where $p$ is a positive constant. The motion takes place in an unusual medium that resists radial motion but not tangential motion, so there is an additional radial force $-k \dot{r} / r^{2}$ where $k$ is a positive constant. Show that the particle's motion lies in a plane. Using polar coordinates in that plane, show also that its angular momentum $h=r^{2} \dot{\theta}$ is constant.

Obtain the equation of motion

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

where $u=r^{-1}$, and find its general solution assuming that $k /|h|<2$. Show that so long as the motion remains bounded it eventually becomes circular with radius $h^{2} / p$.

Obtain the expression

$E=\frac{1}{2} h^{2}\left(u^{2}+\left(\frac{d u}{d \theta}\right)^{2}\right)-p u$

for the particle's total energy, that is, its kinetic energy plus its electrostatic potential energy. Hence, or otherwise, show that the energy is a decreasing function of time.

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• # Paper 4, Section II, C

A particle of mass $m$ and charge $q$ has position vector $\mathbf{r}(t)$ and moves in a constant, uniform magnetic field $\mathbf{B}$ so that its equation of motion is

$m \ddot{\mathbf{r}}=q \dot{\mathbf{r}} \times \mathbf{B}$

Let $\mathbf{L}=m \mathbf{r} \times \dot{\mathbf{r}}$ be the particle's angular momentum. Show that

$\mathbf{L} \cdot \mathbf{B}+\frac{1}{2} q|\mathbf{r} \times \mathbf{B}|^{2}$

is a constant of the motion. Explain why the kinetic energy $T$ is also constant, and show that it may be written in the form

$T=\frac{1}{2} m \mathbf{u} \cdot\left((\mathbf{u} \cdot \mathbf{v}) \mathbf{v}-r^{2} \ddot{\mathbf{u}}\right)$

where $\mathbf{v}=\dot{\mathbf{r}}, r=|\mathbf{r}|$ and $\mathbf{u}=\mathbf{r} / r$.

[Hint: Consider u $\cdot \dot{\mathbf{u}} .]$

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• # Paper 4 , Section I, E

State the Chinese remainder theorem and Fermat's theorem. Prove that

$p^{4} \equiv 1 \quad(\bmod 240)$

for any prime $p>5$.

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• # Paper 4, Section I, E

(a) Find all integers $x$ and $y$ such that

$6 x+2 y \equiv 3 \quad(\bmod 53) \quad \text { and } \quad 17 x+4 y \equiv 7 \quad(\bmod 53)$

(b) Show that if an integer $n>4$ is composite then $(n-1) ! \equiv 0(\bmod n)$.

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• # Paper 4, Section II, E

What does it mean for a set to be countable? Prove that

(a) if $B$ is countable and $f: A \rightarrow B$ is injective, then $A$ is countable;

(b) if $A$ is countable and $f: A \rightarrow B$ is surjective, then $B$ is countable.

Prove that $\mathbb{N} \times \mathbb{N}$ is countable, and deduce that

(i) if $X$ and $Y$ are countable, then so is $X \times Y$;

(ii) $\mathbb{Q}$ is countable.

Let $\mathcal{C}$ be a collection of circles in the plane such that for each point $a$ on the $x$-axis, there is a circle in $\mathcal{C}$ passing through the point $a$ which has the $x$-axis tangent to the circle at $a$. Show that $\mathcal{C}$ contains a pair of circles that intersect.

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• # Paper 4, Section II, E

State the inclusion-exclusion principle.

Let $n \in \mathbb{N}$. A permutation $\sigma$ of the set $\{1,2,3, \ldots, n\}$ is said to contain a transposition if there exist $i, j$ with $1 \leqslant i such that $\sigma(i)=j$ and $\sigma(j)=i$. Derive a formula for the number, $f(n)$, of permutations which do not contain a transposition, and show that

$\lim _{n \rightarrow \infty} \frac{f(n)}{n !}=e^{-\frac{1}{2}}$

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• # Paper 4, Section II, E

Let $p$ be a prime. A base $p$ expansion of an integer $k$ is an expression

$k=k_{0}+p \cdot k_{1}+p^{2} \cdot k_{2}+\cdots+p^{\ell} \cdot k_{\ell}$

for some natural number $\ell$, with $0 \leqslant k_{i} for $i=0,1, \ldots, \ell$.

(i) Show that the sequence of coefficients $k_{0}, k_{1}, k_{2}, \ldots, k_{\ell}$ appearing in a base $p$ expansion of $k$ is unique, up to extending the sequence by zeroes.

(ii) Show that

$\left(\begin{array}{l} p \\ j \end{array}\right) \equiv 0 \quad(\bmod p), \quad 0

and hence, by considering the polynomial $(1+x)^{p}$ or otherwise, deduce that

$\left(\begin{array}{c} p^{i} \\ j \end{array}\right) \equiv 0 \quad(\bmod p), \quad 0

(iii) If $n_{0}+p \cdot n_{1}+p^{2} \cdot n_{2}+\cdots+p^{\ell} \cdot n_{\ell}$ is a base $p$ expansion of $n$, then, by considering the polynomial $(1+x)^{n}$ or otherwise, show that

$\left(\begin{array}{l} n \\ k \end{array}\right) \equiv\left(\begin{array}{l} n_{0} \\ k_{0} \end{array}\right)\left(\begin{array}{l} n_{1} \\ k_{1} \end{array}\right) \cdots\left(\begin{array}{l} n_{\ell} \\ k_{\ell} \end{array}\right) \quad(\bmod p)$

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• # Paper 4, Section II, E

(i) Let $\sim$ be an equivalence relation on a set $X$. What is an equivalence class of $\sim$ ? What is a partition of $X ?$ Prove that the equivalence classes of $\sim$ form a partition of $X$.

(ii) Let $\sim$ be the relation on the natural numbers $\mathbb{N}=\{1,2,3, \ldots\}$ defined by

$m \sim n \Longleftrightarrow \exists a, b \in \mathbb{N} \text { such that } m \text { divides } n^{a} \text { and } n \text { divides } m^{b} .$

Show that $\sim$ is an equivalence relation, and show that it has infinitely many equivalence classes, all but one of which are infinite.

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