Part IA, 2010, Paper 4

# Part IA, 2010, Paper 4

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Paper 4 , Section II, B

commentA sphere of uniform density has mass $m$ and radius $a$. Find its moment of inertia about an axis through its centre.

A marble of uniform density is released from rest on a plane inclined at an angle $\alpha$ to the horizontal. Let the time taken for the marble to travel a distance $\ell$ down the plane be: (i) $t_{1}$ if the plane is perfectly smooth; or (ii) $t_{2}$ if the plane is rough and the marble rolls without slipping.

Explain, with a clear discussion of the forces acting on the marble, whether or not its energy is conserved in each of the cases (i) and (ii). Show that $t_{1} / t_{2}=\sqrt{5 / 7}$.

Suppose that the original marble is replaced by a new one with the same mass and radius but with a hollow centre, so that its moment of inertia is $\lambda m a^{2}$ for some constant $\lambda$. What is the new value for $t_{1} / t_{2}$ ?

Paper 4, Section $I$, B

commentLet $S$ be an inertial frame with coordinates $(t, x)$ in two-dimensional spacetime. Write down the Lorentz transformation giving the coordinates $\left(t^{\prime}, x^{\prime}\right)$ in a second inertial frame $S^{\prime}$ moving with velocity $v$ relative to $S$. If a particle has constant velocity $u$ in $S$, find its velocity $u^{\prime}$ in $S^{\prime}$. Given that $|u|<c$ and $|v|<c$, show that $\left|u^{\prime}\right|<c$.

Paper 4, Section I, B

commentA particle of mass $m$ and charge $q$ moves with trajectory $\mathbf{r}(t)$ in a constant magnetic field $\mathbf{B}=B \hat{\mathbf{z}}$. Write down the Lorentz force on the particle and use Newton's Second Law to deduce that

$\dot{\mathbf{r}}-\omega \mathbf{r} \times \hat{\mathbf{z}}=\mathbf{c}$

where $\mathbf{c}$ is a constant vector and $\omega$ is to be determined. Find $\mathbf{c}$ and hence $\mathbf{r}(t)$ for the initial conditions

$\mathbf{r}(0)=a \hat{\mathbf{x}} \quad \text { and } \quad \dot{\mathbf{r}}(0)=u \hat{\mathbf{y}}+v \hat{\mathbf{z}}$

where $a, u$ and $v$ are constants. Sketch the particle's trajectory in the case $a \omega+u=0$.

[Unit vectors $\hat{\mathbf{x}}, \hat{\mathbf{y}}, \hat{\mathbf{z}}$ correspond to a set of Cartesian coordinates. ]

Paper 4, Section II, B

commentA particle $A$ of rest mass $m$ is fired at an identical particle $B$ which is stationary in the laboratory. On impact, $A$ and $B$ annihilate and produce two massless photons whose energies are equal. Assuming conservation of four-momentum, show that the angle $\theta$ between the photon trajectories is given by

$\cos \theta=\frac{E-3 m c^{2}}{E+m c^{2}}$

where $E$ is the relativistic energy of $A$.

Let $v$ be the speed of the incident particle $A$. For what value of $v / c$ will the photons move in perpendicular directions? If $v$ is very small compared with $c$, show that

$\theta \approx \pi-v / c$

[All quantities referred to are measured in the laboratory frame.]

Paper 4, Section II, B

commentConsider a set of particles with position vectors $\mathbf{r}_{i}(t)$ and masses $m_{i}$, where $i=1,2, \ldots, N$. Particle $i$ experiences an external force $\mathbf{F}_{i}$ and an internal force $\mathbf{F}_{i j}$ from particle $j$, for each $j \neq i$. Stating clearly any assumptions you need, show that

$\frac{d \mathbf{P}}{d t}=\mathbf{F} \quad \text { and } \quad \frac{d \mathbf{L}}{d t}=\mathbf{G}$

where $\mathbf{P}$ is the total momentum, $\mathbf{F}$ is the total external force, $\mathbf{L}$ is the total angular momentum about a fixed point $\mathbf{a}$, and $\mathbf{G}$ is the total external torque about $\mathbf{a}$.

Does the result $\frac{d \mathbf{L}}{d t}=\mathbf{G}$ still hold if the fixed point $\mathbf{a}$ is replaced by the centre of mass of the system? Justify your answer.

Suppose now that the external force on particle $i$ is $-k \frac{d \mathbf{r}_{i}}{d t}$ and that all the particles have the same mass $m$. Show that

$\mathbf{L}(t)=\mathbf{L}(0) e^{-k t / m}$

Paper 4, Section II, B

commentA particle of unit mass moves in a plane with polar coordinates $(r, \theta)$ and components of acceleration $\left(\ddot{r}-r \dot{\theta}^{2}, r \ddot{\theta}+2 \dot{r} \dot{\theta}\right)$. The particle experiences a force corresponding to a potential $-Q / r$. Show that

$E=\frac{1}{2} \dot{r}^{2}+U(r) \quad \text { and } \quad h=r^{2} \dot{\theta}$

are constants of the motion, where

$U(r)=\frac{h^{2}}{2 r^{2}}-\frac{Q}{r}$

Sketch the graph of $U(r)$ in the cases $Q>0$ and $Q<0$.

(a) Assuming $Q>0$ and $h>0$, for what range of values of $E$ do bounded orbits exist? Find the minimum and maximum distances from the origin, $r_{\min }$ and $r_{\max }$, on such an orbit and show that

$r_{\min }+r_{\max }=\frac{Q}{|E|} .$

Prove that the minimum and maximum values of the particle's speed, $v_{\min }$ and $v_{\max }$, obey

$v_{\min }+v_{\max }=\frac{2 Q}{h}$

(b) Now consider trajectories with $E>0$ and $Q$ of either sign. Find the distance of closest approach, $r_{\min }$, in terms of the impact parameter, $b$, and $v_{\infty}$, the limiting value of the speed as $r \rightarrow \infty$. Deduce that if $b \ll|Q| / v_{\infty}^{2}$ then, to leading order,

$r_{\min } \approx \frac{2|Q|}{v_{\infty}^{2}} \text { for } Q<0, \quad r_{\min } \approx \frac{b^{2} v_{\infty}^{2}}{2 Q} \text { for } Q>0$

Paper 4 , Section II, E

commentWhat does it mean for a set to be countable ?

Show that $\mathbb{Q}$ is countable, but $\mathbb{R}$ is not. Show also that the union of two countable sets is countable.

A subset $A$ of $\mathbb{R}$ has the property that, given $\epsilon>0$ and $x \in \mathbb{R}$, there exist reals $a, b$ with $a \in A$ and $b \notin A$ with $|x-a|<\epsilon$ and $|x-b|<\epsilon$. Can $A$ be countable ? Can $A$ be uncountable ? Justify your answers.

A subset $B$ of $\mathbb{R}$ has the property that given $b \in B$ there exists $\epsilon>0$ such that if $0<|b-x|<\epsilon$ for some $x \in \mathbb{R}$, then $x \notin B$. Is $B$ countable ? Justify your answer.

Paper 4, Section I, $2 E$

comment(a) Let $r$ be a real root of the polynomial $f(x)=x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0}$, with integer coefficients $a_{i}$ and leading coefficient 1 . Show that if $r$ is rational, then $r$ is an integer.

(b) Write down a series for $e$. By considering $q ! e$ for every natural number $q$, show that $e$ is irrational.

Paper 4, Section I, E

comment(a) Find the smallest residue $x$ which equals $28 ! 13^{28}(\bmod 31)$.

[You may use any standard theorems provided you state them correctly.]

(b) Find all integers $x$ which satisfy the system of congruences

$\begin{aligned} x & \equiv 1(\bmod 2) \\ 2 x & \equiv 1(\bmod 3) \\ 2 x & \equiv 4(\bmod 10) \\ x & \equiv 10(\bmod 67) \end{aligned}$

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

commentState and prove Fermat's Little Theorem.

Let $p$ be an odd prime. If $p \neq 5$, show that $p$ divides $10^{n}-1$ for infinitely many natural numbers $n$.

Hence show that $p$ divides infinitely many of the integers

$5,55, \quad 555, \quad 5555, \quad \ldots .$

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

comment(a) Let $A, B$ be finite non-empty sets, with $|A|=a,|B|=b$. Show that there are $b^{a}$ mappings from $A$ to $B$. How many of these are injective ?

(b) State the Inclusion-Exclusion principle.

(c) Prove that the number of surjective mappings from a set of size $n$ onto a set of size $k$ is

$\sum_{i=0}^{k}(-1)^{i}\left(\begin{array}{c} k \\ i \end{array}\right)(k-i)^{n} \quad \text { for } n \geqslant k \geqslant 1$

Deduce that

$n !=\sum_{i=0}^{n}(-1)^{i}\left(\begin{array}{c} n \\ i \end{array}\right)(n-i)^{n}$

Paper 4, Section II, E

commentThe Fibonacci numbers $F_{n}$ are defined for all natural numbers $n$ by the rules

$F_{1}=1, \quad F_{2}=1, \quad F_{n}=F_{n-1}+F_{n-2} \quad \text { for } n \geqslant 3$

Prove by induction on $k$ that, for any $n$,

$F_{n+k}=F_{k} F_{n+1}+F_{k-1} F_{n} \text { for all } k \geqslant 2 \text {. }$

Deduce that

$F_{2 n}=F_{n}\left(F_{n+1}+F_{n-1}\right) \quad \text { for all } n \geqslant 2$

Put $L_{1}=1$ and $L_{n}=F_{n+1}+F_{n-1}$ for $n>1$. Show that these (Lucas) numbers $L_{n}$ satisfy

$L_{1}=1, \quad L_{2}=3, \quad L_{n}=L_{n-1}+L_{n-2} \quad \text { for } n \geqslant 3$

Show also that, for all $n$, the greatest common divisor $\left(F_{n}, F_{n+1}\right)$ is 1 , and that the greatest common divisor $\left(F_{n}, L_{n}\right)$ is at most 2 .