Part IA, 2011, Paper 4

Dynamics and Relativity

• Paper 4, Section I, B

  Dynamics and Relativity | Part IA, 2011

  Inertial frames $S$ and $S^{\prime}$ in two-dimensional space-time have coordinates $(x, t)$ and $\left(x^{\prime}, t^{\prime}\right)$, respectively. These coordinates are related by a Lorentz transformation with $v$ the velocity of $S^{\prime}$ relative to $S$. Show that if $x_{\pm}=x \pm c t$ and $x_{\pm}^{\prime}=x^{\prime} \pm c t^{\prime}$ then the Lorentz transformation can be expressed in the form

  $$x_{+}^{\prime}=\lambda(v) x_{+} \quad \text { and } \quad x_{-}^{\prime}=\lambda(-v) x_{-}, \quad \text { where } \quad \lambda(v)=\left(\frac{c-v}{c+v}\right)^{1 / 2} .$$

  Deduce that $x^{2}-c^{2} t^{2}=x^{\prime 2}-c^{2} t^{\prime 2}$.

  Use the form $(*)$ to verify that successive Lorentz transformations with velocities $v_{1}$ and $v_{2}$ result in another Lorentz transformation with velocity $v_{3}$, to be determined in terms of $v_{1}$ and $v_{2}$.

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• Paper 4, Section I, B

  Dynamics and Relativity | Part IA, 2011

  The motion of a planet in the gravitational field of a star of mass $M$ obeys

  $$\frac{\mathrm{d}^{2} r}{\mathrm{~d} t^{2}}-\frac{h^{2}}{r^{3}}=-\frac{G M}{r^{2}}, \quad r^{2} \frac{\mathrm{d} \theta}{\mathrm{d} t}=h$$

  where $r(t)$ and $\theta(t)$ are polar coordinates in a plane and $h$ is a constant. Explain one of Kepler's Laws by giving a geometrical interpretation of $h$.

  Show that circular orbits are possible, and derive another of Kepler's Laws relating the radius $a$ and the period $T$ of such an orbit. Show that any circular orbit is stable under small perturbations that leave $h$ unchanged.

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• Paper 4, Section II, B

  Dynamics and Relativity | Part IA, 2011

  (a) Write down the relativistic energy $E$ of a particle of rest mass $m$ and speed $v$. Find the approximate form for $E$ when $v$ is small compared to $c$, keeping all terms up to order $(v / c)^{2}$. What new physical idea (when compared to Newtonian Dynamics) is revealed in this approximation?

  (b) A particle of rest mass $m$ is fired at an identical particle which is at rest in the laboratory frame. Let $E$ be the relativistic energy and $v$ the speed of the incident particle in this frame. After the collision, there are $N$ particles in total, each with rest mass $m$. Assuming that four-momentum is conserved, find a lower bound on $E$ and hence show that

  $$v \geqslant \frac{N\left(N^{2}-4\right)^{1 / 2}}{N^{2}-2} c$$

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• Paper 4, Section II, B

  Dynamics and Relativity | Part IA, 2011

  A rocket carries equipment to collect samples from a stationary cloud of cosmic dust. The rocket moves in a straight line, burning fuel and ejecting gas at constant speed $u$ relative to itself. Let $v(t)$ be the speed of the rocket, $M(t)$ its total mass, including fuel and any dust collected, and $m(t)$ the total mass of gas that has been ejected. Show that

  $$M \frac{\mathrm{d} v}{\mathrm{~d} t}+v \frac{\mathrm{d} M}{\mathrm{~d} t}+(v-u) \frac{\mathrm{d} m}{\mathrm{dt}}=0$$

  assuming that all external forces are negligible.

  (a) If no dust is collected and the rocket starts from rest with mass $M_{0}$, deduce that

  $$v=u \log \left(M_{0} / M\right)$$

  (b) If cosmic dust is collected at a constant rate of $\alpha$ units of mass per unit time and fuel is consumed at a constant rate $\mathrm{d} m / \mathrm{d} t=\beta$, show that, with the same initial conditions as in (a),

  $$v=\frac{u \beta}{\alpha}\left(1-\left(M / M_{0}\right)^{\alpha /(\beta-\alpha)}\right)$$

  Verify that the solution in (a) is recovered in the limit $\alpha \rightarrow 0$.

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• Paper 4, Section II, B

  Dynamics and Relativity | Part IA, 2011

  The trajectory of a particle $\mathbf{r}(t)$ is observed in a frame $S$ which rotates with constant angular velocity $\omega$ relative to an inertial frame $I$. Given that the time derivative in $I$ of any vector $\mathbf{u}$ is

  $$\left(\frac{\mathrm{d} \mathbf{u}}{\mathrm{d} t}\right)_{I}=\dot{\mathbf{u}}+\boldsymbol{\omega} \times \mathbf{u},$$

  where a dot denotes a time derivative in $S$, show that

  $$m \ddot{\mathbf{r}}=\mathbf{F}-2 m \boldsymbol{\omega} \times \dot{\mathbf{r}}-m \boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{r}),$$

  where $\mathbf{F}$ is the force on the particle and $m$ is its mass.

  Let $S$ be the frame that rotates with the Earth. Assume that the Earth is a sphere of radius $R$. Let $P$ be a point on its surface at latitude $\pi / 2-\theta$, and define vertical to be the direction normal to the Earth's surface at $P$.

  (a) A particle at $P$ is released from rest in $S$ and is acted on only by gravity. Show that its initial acceleration makes an angle with the vertical of approximately

  $$\frac{\omega^{2} R}{g} \sin \theta \cos \theta$$

  working to lowest non-trivial order in $\omega$.

  (b) Now consider a particle fired vertically upwards from $P$ with speed $v$. Assuming that terms of order $\omega^{2}$ and higher can be neglected, show that it falls back to Earth under gravity at a distance

  $$\frac{4}{3} \frac{\omega v^{3}}{g^{2}} \sin \theta$$

  from $P$. [You may neglect the curvature of the Earth's surface and the vertical variation of gravity.]

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• Paper 4, Section II, B

  Dynamics and Relativity | Part IA, 2011

  A particle with mass $m$ and position $\mathbf{r}(t)$ is subject to a force

  $$\mathbf{F}=\mathbf{A}(\mathbf{r})+\dot{\mathbf{r}} \times \mathbf{B}(\mathbf{r})$$

  (a) Suppose that $\mathbf{A}=-\nabla \phi$. Show that

  $$E=\frac{1}{2} m \dot{\mathbf{r}}^{2}+\phi(\mathbf{r})$$

  is constant, and interpret this result, explaining why the field $\mathbf{B}$ plays no role.

  (b) Suppose, in addition, that $\mathbf{B}=-\nabla \psi$ and that both $\phi$ and $\psi$ depend only on $r=|\mathbf{r}|$. Show that

  $$\mathbf{L}=m \mathbf{r} \times \dot{\mathbf{r}}-\psi \mathbf{r}$$

  is independent of time if $\psi(r)=\mu / r$, for any constant $\mu$.

  (c) Now specialise further to the case $\psi=0$. Explain why the result in (b) implies that the motion of the particle is confined to a plane. Show also that

  $$\mathbf{K}=\mathbf{L} \times \dot{\mathbf{r}}-\phi \mathbf{r}$$

  is constant provided $\phi(r)$ takes a certain form, to be determined.

  [ Recall that $\mathbf{r} \cdot \dot{\mathbf{r}}=r \dot{r}$ and that if $f$ depends only on $r=|\mathbf{r}|$ then $\left.\boldsymbol{\nabla} f=f^{\prime}(r) \hat{\mathbf{r}} .\right]$

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Numbers and Sets

• Paper 4, Section I, E

  Numbers and Sets | Part IA, 2011

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

  Let $\sim$ be the relation on the positive integers defined by $x \sim y$ if either $x$ divides $y$ or $y$ divides $x$. Is $\sim$ an equivalence relation? Justify your answer.

  Write down an equivalence relation on the positive integers that has exactly four equivalence classes, of which two are infinite and two are finite.

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• Paper 4, Section I, E

  Numbers and Sets | Part IA, 2011

  What does it mean to say that a function $f: X \rightarrow Y$ has an inverse? Show that a function has an inverse if and only if it is a bijection.

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

  (i) If $f$ and $g$ are bijections then $f \circ g$ is a bijection.

  (ii) If $f \circ g$ is a bijection then $f$ and $g$ are bijections.

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• Paper 4, Section II, $7 \mathrm{E}$

  Numbers and Sets | Part IA, 2011

  Define the binomial coefficient $\left(\begin{array}{l}n \\ i\end{array}\right)$, where $n$ is a positive integer and $i$ is an integer with $0 \leqslant i \leqslant n$. Arguing from your definition, show that $\sum_{i=0}^{n}\left(\begin{array}{l}n \\ i\end{array}\right)=2^{n}$.

  Prove the binomial theorem, that $(1+x)^{n}=\sum_{i=0}^{n}\left(\begin{array}{l}n \\ i\end{array}\right) x^{i}$ for any real number $x$.

  By differentiating this expression, or otherwise, evaluate $\sum_{i=0}^{n} i\left(\begin{array}{l}n \\ i\end{array}\right)$ and $\sum_{i=0}^{n} i^{2}\left(\begin{array}{l}n \\ i\end{array}\right)$. By considering the identity $(1+x)^{n}(1+x)^{n}=(1+x)^{2 n}$, or otherwise, show that

  $$\sum_{i=0}^{n}\left(\begin{array}{l} n \\ i \end{array}\right)^{2}=\left(\begin{array}{c} 2 n \\ n \end{array}\right)$$

  Show that $\sum_{i=0}^{n} i\left(\begin{array}{c}n \\ i\end{array}\right)^{2}=\frac{n}{2}\left(\begin{array}{c}2 n \\ n\end{array}\right)$

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• Paper 4, Section II, E

  Numbers and Sets | Part IA, 2011

  Show that, for any set $X$, there is no surjection from $X$ to the power-set of $X$.

  Show that there exists an injection from $\mathbb{R}^{2}$ to $\mathbb{R}$.

  Let $A$ be a subset of $\mathbb{R}^{2}$. A section of $A$ is a subset of $\mathbb{R}$ of the form

  $$\{t \in \mathbb{R}: a+t b \in A\},$$

  where $a \in \mathbb{R}^{2}$ and $b \in \mathbb{R}^{2}$ with $b \neq 0$. Prove that there does not exist a set $A \subset \mathbb{R}^{2}$ such that every set $S \subset \mathbb{R}$ is a section of $A$.

  Does there exist a set $A \subset \mathbb{R}^{2}$ such that every countable set $S \subset \mathbb{R}$ is a section of $A ? \quad$ [There is no requirement that every section of $A$ should be countable.] Justify your answer.

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• Paper 4, Section II, E

  Numbers and Sets | Part IA, 2011

  State Fermat's Theorem and Wilson's Theorem.

  Let $p$ be a prime.

  (a) Show that if $p \equiv 3(\bmod 4)$ then the equation $x^{2} \equiv-1(\bmod p)$ has no solution.

  (b) By considering $\left(\frac{p-1}{2}\right)$ !, or otherwise, show that if $p \equiv 1(\bmod 4)$ then the equation $x^{2} \equiv-1(\bmod p)$ does have a solution.

  (c) Show that if $p \equiv 2(\bmod 3)$ then the equation $x^{3} \equiv-1(\bmod p)$ has no solution other than $-1(\bmod p)$.

  (d) Using the fact that $14^{2} \equiv-3(\bmod 199)$, find a solution of $x^{3} \equiv-1(\bmod 199)$ that is not $-1(\bmod 199)$.

  [Hint: how are the complex numbers $\sqrt{-3}$ and $\sqrt[3]{-1}$ related?]

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• Paper 4, Section II, E

  Numbers and Sets | Part IA, 2011

  (a) What is the highest common factor of two positive integers $a$ and $b$ ? Show that the highest common factor may always be expressed in the form $\lambda a+\mu b$, where $\lambda$ and $\mu$ are integers.

  Which positive integers $n$ have the property that, for any positive integers $a$ and $b$, if $n$ divides $a b$ then $n$ divides $a$ or $n$ divides $b$ ? Justify your answer.

  Let $a, b, c, d$ be distinct prime numbers. Explain carefully why $a b$ cannot equal $c d$.

  [No form of the Fundamental Theorem of Arithmetic may be assumed without proof.]

  (b) Now let $S$ be the set of positive integers that are congruent to 1 mod 10 . We say that $x \in S$ is irreducible if $x>1$ and whenever $a, b \in S$ satisfy $a b=x$ then $a=1$ or $b=1$. Do there exist distinct irreducibles $a, b, c, d$ with $a b=c d ?$

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