Part IA, 2006, Paper 4

# Part IA, 2006, Paper 4

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4.I.3C

commentA car is at rest on a horizontal surface. The engine is switched on and suddenly sets the wheels spinning at a constant angular velocity $\Omega$. The wheels have radius $r$ and the coefficient of friction between the ground and the surface of the wheels is $\mu$. Calculate the time $T$ when the wheels start rolling without slipping. If the car is started on an upward slope in a similar manner, explain whether $T$ is increased or decreased relative to the case where the car starts on a horizontal surface.

4.I.4C

commentFor the dynamical system

$\ddot{x}=-\sin x$

find the stable and unstable fixed points and the equation determining the separatrix. Sketch the phase diagram. If the system starts on the separatrix at $x=0$, write down an integral determining the time taken for the velocity $\dot{x}$ to reach zero. Show that the integral is infinite.

4.II.10C

commentA particle of mass $m$ bounces back and forth between two walls of mass $M$ moving towards each other in one dimension. The walls are separated by a distance $\ell(t)$. The wall on the left has velocity $+V(t)$ and the wall on the right has velocity $-V(t)$. The particle has speed $v(t)$. Friction is negligible and the particle-wall collisions are elastic.

Consider a collision between the particle and the wall on the right. Show that the centre-of-mass velocity of the particle-wall system is $v_{\mathrm{cm}}=(m v-M V) /(m+M)$. Calculate the particle's speed following the collision.

Assume that the particle is much lighter than the walls, i.e., $m \ll M$. Show that the particle's speed increases by approximately $2 V$ every time it collides with a wall.

Assume also that $v \gg V$ (so that particle-wall collisions are frequent) and that the velocities of the two walls remain nearly equal and opposite. Show that in a time interval $\Delta t$, over which the change in $V$ is negligible, the wall separation changes by $\Delta \ell \approx-2 V \Delta t$. Show that the number of particle-wall collisions during $\Delta t$ is approximately $v \Delta t / \ell$ and that the particle's speed increases by $\Delta v \approx-(\Delta \ell / \ell) v$ during this time interval.

Hence show that under the given conditions the particle speed $v$ is approximately proportional to $\ell^{-1}$.

4.II.11C

commentTwo light, rigid rods of length $2 \ell$ have a mass $m$ attached to each end. Both are free to move in two dimensions. The two rods are placed so that their two ends are located at $(-d,+2 \ell),(-d, 0)$, and $(+d, 0),(+d,-2 \ell)$ respectively, where $d$ is positive. They are set in motion with no rotation, with centre-of-mass velocities $(+V, 0)$ and $(-V, 0)$, so that the lower mass on the first rod collides head on with the upper mass on the second rod at the origin $(0,0)$. [You may assume that the impulse is directed along the $x$-axis.]

Assuming the collision is elastic, calculate the centre of-mass velocity $\boldsymbol{v}$ and the angular velocity $\boldsymbol{\omega}$ of each rod immediately after the collision.

Assuming a coefficient of restitution $e$, compute $\boldsymbol{v}$ and $\boldsymbol{\omega}$ for each rod after the collision.

4.II.12C

commentA particle of mass $m$ and charge $q>0$ moves in a time-dependent magnetic field $\mathbf{B}=\left(0,0, B_{z}(t)\right)$.

Write down the equations of motion governing the particle's $x, y$ and $z$ coordinates.

Show that the speed of the particle in the $(x, y)$ plane, $V=\sqrt{\dot{x}^{2}+\dot{y}^{2}}$, is a constant.

Show that the general solution of the equations of motion is

$\begin{aligned} &x(t)=x_{0}+V \int_{0}^{t} d t^{\prime} \cos \left(-\int_{0}^{t^{\prime}} d t^{\prime \prime} q \frac{B_{z}\left(t^{\prime \prime}\right)}{m}+\phi\right) \\ &y(t)=y_{0}+V \int_{0}^{t} d t^{\prime} \sin \left(-\int_{0}^{t^{\prime}} d t^{\prime \prime} q \frac{B_{z}\left(t^{\prime \prime}\right)}{m}+\phi\right) \\ &z(t)=z_{0}+v_{z} t \end{aligned}$

and interpret each of the six constants of integration, $x_{0}, y_{0}, z_{0}, v_{z}, V$ and $\phi$. [Hint: Solve the equations for the particle's velocity in cylindrical polars.]

Let $B_{z}(t)=\beta t$, where $\beta$ is a positive constant. Assuming that $x_{0}=y_{0}=z_{0}=$ $v_{z}=\phi=0$ and $V=1$, calculate the position of the particle in the limit $t \rightarrow \infty$ (you may assume this limit exists). [Hint: You may use the results $\int_{0}^{\infty} d x \cos \left(x^{2}\right)=\int_{0}^{\infty} d x \sin \left(x^{2}\right)=$ $\sqrt{\pi / 8} .]$

4.II.9C

commentA motorcycle of mass $M$ moves on a bowl-shaped surface specified by its height $h(r)$ where $r=\sqrt{x^{2}+y^{2}}$ is the radius in cylindrical polar coordinates $(r, \phi, z)$. The torque exerted by the motorcycle engine on the rear wheel results in a force $\mathbf{F}(t)$ pushing the motorcycle forward. Assuming $\mathbf{F}(t)$ is directed along the motorcycle's velocity and that the motorcycle's vertical velocity and acceleration are small, show that the motion is described by

$\begin{aligned} \ddot{r}-r \dot{\phi}^{2} &=-g \frac{d h}{d r}+\frac{F(t)}{M} \frac{\dot{r}}{\sqrt{\dot{r}^{2}+r^{2} \dot{\phi}^{2}}} \\ r \ddot{\phi}+2 \dot{r} \dot{\phi} &=\frac{F(t)}{M} \frac{r \dot{\phi}}{\sqrt{\dot{r}^{2}+r^{2} \dot{\phi}^{2}}} \end{aligned}$

where dots denote time derivatives, $F(t)=|\mathbf{F}(t)|$ and $g$ is the acceleration due to gravity.

The motorcycle rider can adjust $F(t)$ to produce the desired trajectory. If the rider wants to move on a curve $r(\phi)$, show that $\phi(t)$ must obey

$\dot{\phi}^{2}=g \frac{d h}{d r} /\left(r+\frac{2}{r}\left(\frac{d r}{d \phi}\right)^{2}-\frac{d^{2} r}{d \phi^{2}}\right)$

Now assume that $h(r)=r^{2} / \ell$, with $\ell$ a constant, and $r(\phi)=\epsilon \phi$ with $\epsilon$ a positive constant, and $0 \leqslant \phi<\infty$ so that the desired trajectory is a spiral curve. Assuming that $\phi(t)$ tends to infinity as $t$ tends to infinity, show that $\dot{\phi}(t)$ tends to $\sqrt{2 g / \ell}$ and $F(t)$ tends to $4 \epsilon M g / \ell$ as $t$ tends to infinity.

4.I.1E

commentExplain what is meant by a prime number.

By considering numbers of the form $6 p_{1} p_{2} \cdots p_{n}-1$, show that there are infinitely many prime numbers of the form $6 k-1$.

By considering numbers of the form $\left(2 p_{1} p_{2} \cdots p_{n}\right)^{2}+3$, show that there are infinitely many prime numbers of the form $6 k+1$. [You may assume the result that, for a prime $p>3$, the congruence $x^{2} \equiv-3(\bmod p)$ is soluble only if $\left.p \equiv 1(\bmod 6) .\right]$

4.I.2E

commentDefine the binomial coefficient $\left(\begin{array}{l}n \\ r\end{array}\right)$ and prove that

$\left(\begin{array}{c} n+1 \\ r \end{array}\right)=\left(\begin{array}{c} n \\ r \end{array}\right)+\left(\begin{array}{c} n \\ r-1 \end{array}\right) \quad \text { for } 0<r \leqslant n$

Show also that if $p$ is prime then $\left(\begin{array}{l}p \\ r\end{array}\right)$ is divisible by $p$ for $0<r<p$.

Deduce that if $0 \leqslant k<p$ and $0 \leqslant r \leqslant k$ then

$\left(\begin{array}{c} p+k \\ r \end{array}\right) \equiv\left(\begin{array}{c} k \\ r \end{array}\right) \quad(\bmod p) .$

4.II.5E

commentExplain what is meant by an equivalence relation on a set $A$.

If $R$ and $S$ are two equivalence relations on the same set $A$, we define

$R \circ S=\{(x, z) \in A \times A:$ there exists $y \in A$ such that $(x, y) \in R$ and $(y, z) \in S\} .$

Show that the following conditions are equivalent:

(i) $R \circ S$ is a symmetric relation on $A$;

(ii) $R \circ S$ is a transitive relation on $A$;

(iii) $S \circ R \subseteq R \circ S$;

(iv) $R \circ S$ is the unique smallest equivalence relation on $A$ containing both $R$ and $S$.

Show also that these conditions hold if $A=\mathbb{Z}$ and $R$ and $S$ are the relations of congruence modulo $m$ and modulo $n$, for some positive integers $m$ and $n$.

4.II.6E

commentState and prove the Inclusion-Exclusion Principle.

A permutation $\sigma$ of $\{1,2, \ldots, n\}$ is called a derangement if $\sigma(j) \neq j$ for every $j \leqslant n$. Use the Inclusion-Exclusion Principle to find a formula for the number $f(n)$ of derangements of $\{1,2, \ldots, n\}$. Show also that $f(n) / n$ ! converges to $1 / e$ as $n \rightarrow \infty$.

4.II.7E

commentState and prove Fermat's Little Theorem.

An odd number $n$ is called a Carmichael number if it is not prime, but every positive integer $a$ satisfies $a^{n} \equiv a(\bmod n)$. Show that a Carmichael number cannot be divisible by the square of a prime. Show also that a product of two distinct odd primes cannot be a Carmichael number, and that a product of three distinct odd primes $p, q, r$ is a Carmichael number if and only if $p-1$ divides $q r-1, q-1$ divides $p r-1$ and $r-1$ divides $p q-1$. Deduce that 1729 is a Carmichael number.

[You may assume the result that, for any prime $p$, there exists a number g prime to $p$ such that the congruence $g^{d} \equiv 1(\bmod p)$ holds only when $d$ is a multiple of $p-1$. The prime factors of 1729 are 7,13 and 19.]

4.II.8E

commentExplain what it means for a set to be countable. Prove that a countable union of countable sets is countable, and that the set of all subsets of $\mathbb{N}$ is uncountable.

A function $f: \mathbb{N} \rightarrow \mathbb{N}$ is said to be increasing if $f(m) \leqslant f(n)$ whenever $m \leqslant n$, and decreasing if $f(m) \geqslant f(n)$ whenever $m \leqslant n$. Show that the set of all increasing functions $\mathbb{N} \rightarrow \mathbb{N}$ is uncountable, but that the set of decreasing functions is countable.

[Standard results on countability, other than those you are asked to prove, may be assumed.]