Part IA, 2016, Paper 4

# Part IA, 2016, Paper 4

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Paper 4, Section $I$, B

commentThe radial equation of motion of a particle moving under the influence of a central force is

$\ddot{r}-\frac{h^{2}}{r^{3}}=-k r^{n}$

where $h$ is the angular momentum per unit mass of the particle, $n$ is a constant, and $k$ is a positive constant.

Show that circular orbits with $r=a$ are possible for any positive value of $a$, and that they are stable to small perturbations that leave $h$ unchanged if $n>-3$.

Paper 4, Section I, B

commentWith the help of definitions or equations of your choice, determine the dimensions, in terms of mass $(M)$, length $(L)$, time $(T)$ and charge $(Q)$, of the following quantities:

(i) force;

(ii) moment of a force (i.e. torque);

(iii) energy;

(iv) Newton's gravitational constant $G$;

(v) electric field $\mathbf{E}$;

(vi) magnetic field $\mathbf{B}$;

(vii) the vacuum permittivity $\epsilon_{0}$.

Paper 4, Section II, B

commentState what the vectors $\mathbf{a}, \mathbf{r}, \mathbf{v}$ and $\boldsymbol{\omega}$ represent in the following equation:

$\mathbf{a}=\mathbf{g}-2 \boldsymbol{\omega} \times \mathbf{v}-\boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{r})$

where $\mathbf{g}$ is the acceleration due to gravity.

Assume that the radius of the Earth is $6 \times 10^{6} \mathrm{~m}$, that $|\mathrm{g}|=10 \mathrm{~ms}^{-2}$, and that there are $9 \times 10^{4}$ seconds in a day. Use these data to determine roughly the order of magnitude of each term on the right hand side of $(*)$ in the case of a particle dropped from a point at height $20 \mathrm{~m}$ above the surface of the Earth.

Taking again $|\mathbf{g}|=10 \mathrm{~ms}^{-2}$, find the time $T$ of the particle's fall in the absence of rotation.

Use a suitable approximation scheme to show that

$\mathbf{R} \approx \mathbf{R}_{0}-\frac{1}{3} \boldsymbol{\omega} \times \mathbf{g} T^{3}-\frac{1}{2} \boldsymbol{\omega} \times\left(\boldsymbol{\omega} \times \mathbf{R}_{0}\right) T^{2},$

where $\mathbf{R}$ is the position vector of the point at which the particle lands, and $\mathbf{R}_{0}$ is the position vector of the point at which the particle would have landed in the absence of rotation.

The particle is dropped at latitude $45^{\circ}$. Find expressions for the approximate northerly and easterly displacements of $\mathbf{R}$ from $\mathbf{R}_{0}$ in terms of $\omega, g, R_{0}$ (the magnitudes of $\boldsymbol{\omega}, \mathbf{g}$ and $\mathbf{R}_{0}$, respectively), and $T$. You should ignore the curvature of the Earth's surface.

Paper 4, Section II, B

comment(a) Alice travels at constant speed $v$ to Alpha Centauri, which is at distance $d$ from Earth. She then turns around (taking very little time to do so), and returns at speed $v$. Bob stays at home. By how much has Bob aged during the journey? By how much has Alice aged? [No justification is required.]

Briefly explain what is meant by the twin paradox in this context. Why is it not a paradox?

(b) Suppose instead that Alice's world line is given by

$-c^{2} t^{2}+x^{2}=c^{2} t_{0}^{2},$

where $t_{0}$ is a positive constant. Bob stays at home, at $x=\alpha c t_{0}$, where $\alpha>1$. Alice and Bob compare their ages on both occasions when they meet. By how much does Bob age? Show that Alice ages by $2 t_{0} \cosh ^{-1} \alpha$.

Paper 4, Section II, B

commentA particle of unit mass moves with angular momentum $h$ in an attractive central force field of magnitude $\frac{k}{r^{2}}$, where $r$ is the distance from the particle to the centre and $k$ is a constant. You may assume that the equation of its orbit can be written in plane polar coordinates in the form

$r=\frac{\ell}{1+e \cos \theta}$

where $\ell=\frac{h^{2}}{k}$ and $e$ is the eccentricity. Show that the energy of the particle is

$\frac{h^{2}\left(e^{2}-1\right)}{2 \ell^{2}}$

A comet moves in a parabolic orbit about the Sun. When it is at its perihelion, a distance $d$ from the Sun, and moving with speed $V$, it receives an impulse which imparts an additional velocity of magnitude $\alpha V$ directly away from the Sun. Show that the eccentricity of its new orbit is $\sqrt{1+4 \alpha^{2}}$, and sketch the two orbits on the same axes.

Paper 4, Section II, B

comment(a) A rocket, moving non-relativistically, has speed $v(t)$ and mass $m(t)$ at a time $t$ after it was fired. It ejects mass with constant speed $u$ relative to the rocket. Let the total momentum, at time $t$, of the system (rocket and ejected mass) in the direction of the motion of the rocket be $P(t)$. Explain carefully why $P(t)$ can be written in the form

$\tag{*} P(t)=m(t) v(t)-\int_{0}^{t}(v(\tau)-u) \frac{d m(\tau)}{d \tau} d \tau$

If the rocket experiences no external force, show that

$\tag{†} m \frac{d v}{d t}+u \frac{d m}{d t}=0$

Derive the expression corresponding to $(*)$ for the total kinetic energy of the system at time $t$. Show that kinetic energy is not necessarily conserved.

(b) Explain carefully how $(*)$ should be modified for a rocket moving relativistically, given that there are no external forces. Deduce that

$\frac{d(m \gamma v)}{d t}=\left(\frac{v-u}{1-u v / c^{2}}\right) \frac{d(m \gamma)}{d t}$

where $\gamma=\left(1-v^{2} / c^{2}\right)^{-\frac{1}{2}}$ and hence that

$\tag{‡} m \gamma^{2} \frac{d v}{d t}+u \frac{d m}{d t}=0$

(c) Show that $(†)$ and $(‡)$ agree in the limit $c \rightarrow \infty$. Briefly explain the fact that kinetic energy is not conserved for the non-relativistic rocket, but relativistic energy is conserved for the relativistic rocket.

Paper 4, Section I, E

commentExplain the meaning of the phrase least upper bound; state the least upper bound property of the real numbers. Use the least upper bound property to show that a bounded, increasing sequence of real numbers converges.

Suppose that $a_{n}, b_{n} \in \mathbb{R}$ and that $a_{n} \geqslant b_{n}>0$ for all $n$. If $\sum_{n=1}^{\infty} a_{n}$ converges, show that $\sum_{n=1}^{\infty} b_{n}$ converges.

Paper 4, Section I, E

commentFind a pair of integers $x$ and $y$ satisfying $17 x+29 y=1$. What is the smallest positive integer congruent to $17^{138}$ modulo 29 ?

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

commentSuppose that $a, b \in \mathbb{Z}$ and that $b=b_{1} b_{2}$, where $b_{1}$ and $b_{2}$ are relatively prime and greater than 1. Show that there exist unique integers $a_{1}, a_{2}, n \in \mathbb{Z}$ such that $0 \leqslant a_{i}<b_{i}$ and

$\frac{a}{b}=\frac{a_{1}}{b_{1}}+\frac{a_{2}}{b_{2}}+n$

Now let $b=p_{1}^{n_{1}} \cdots p_{k}^{n_{k}}$ be the prime factorization of $b$. Deduce that $\frac{a}{b}$ can be written uniquely in the form

$\frac{a}{b}=\frac{q_{1}}{p_{1}^{n_{1}}}+\cdots+\frac{q_{k}}{p_{k}^{n_{k}}}+n$

where $0 \leqslant q_{i}<p_{i}^{n_{i}}$ and $n \in \mathbb{Z}$. Express $\frac{a}{b}=\frac{1}{315}$ in this form.

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

commentState the inclusion-exclusion principle.

Let $A=\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ be a string of $n$ digits, where $a_{i} \in\{0,1, \ldots, 9\}$. We say that the string $A$ has a run of length $k$ if there is some $j \leqslant n-k+1$ such that either $a_{j+i} \equiv a_{j}+i(\bmod 10)$ for all $0 \leqslant i<k$ or $a_{j+i} \equiv a_{j}-i(\bmod 10)$ for all $0 \leqslant i<k$. For example, the strings

$(\underline{0,1,2}, 8,4,9),(3, \underline{9,8,7}, 4,8) \text { and }(3, \underline{1,0,9}, 4,5)$

all have runs of length 3 (underlined), but no run in $(3,1,2,1,1,2)$ has length $>2$. How many strings of length 6 have a run of length $\geqslant 3$ ?

Paper 4, Section II, 8E

commentDefine the binomial coefficient $\left(\begin{array}{c}n \\ m\end{array}\right)$. Prove directly from your definition that

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

for any complex number $z$.

(a) Using this formula, or otherwise, show that

$\sum_{k=0}^{3 n}(-3)^{k}\left(\begin{array}{l} 6 n \\ 2 k \end{array}\right)=2^{6 n}$

(b) By differentiating, or otherwise, evaluate $\sum_{m=0}^{n} m\left(\begin{array}{c}n \\ m\end{array}\right)$.

Let $S_{r}(n)=\sum_{m=0}^{n}(-1)^{m} m^{r}\left(\begin{array}{c}n \\ m\end{array}\right)$, where $r$ is a non-negative integer. Show that $S_{r}(n)=0$ for $r<n$. Evaluate $S_{n}(n)$.

Paper 4, Section II, E

comment(a) Let $S$ be a set. Show that there is no bijective map from $S$ to the power set of $S$. Let $\mathcal{T}=\left\{\left(x_{n}\right) \mid x_{i} \in\{0,1\}\right.$ for all $\left.i \in \mathbb{N}\right\}$ be the set of sequences with entries in $\{0,1\} .$ Show that $\mathcal{T}$ is uncountable.

(b) Let $A$ be a finite set with more than one element, and let $B$ be a countably infinite set. Determine whether each of the following sets is countable. Justify your answers.

(i) $S_{1}=\{f: A \rightarrow B \mid f$ is injective $\}$.

(ii) $S_{2}=\{g: B \rightarrow A \mid g$ is surjective $\}$.

(iii) $S_{3}=\{h: B \rightarrow B \mid h$ is bijective $\}$.