# Dynamics And Relativity

### Jump to year

Paper 4, Section I, C

commentA rigid body composed of $N$ particles with positions $\mathbf{x}_{i}$, and masses $m_{i}(i=$ $1,2, \ldots, N)$, rotates about the $z$-axis with constant angular speed $\omega$. Show that the body's kinetic energy is $T=\frac{1}{2} I \omega^{2}$, where you should give an expression for the moment of inertia $I$ in terms of the particle masses and positions.

Consider a solid cuboid of uniform density, mass $M$, and dimensions $2 a \times 2 b \times 2 c$. Choose coordinate axes so that the cuboid is described by the points $(x, y, z)$ with $-a \leqslant x \leqslant a,-b \leqslant y \leqslant b$, and $-c \leqslant z \leqslant c$. In terms of $M, a$, $b$, and $c$, find the cuboid's moment of inertia $I$ for rotations about the $z$-axis.

Paper 4, Section I, C

commentA trolley travels with initial speed $v_{0}$ along a frictionless, horizontal, linear track. It slows down by ejecting gas in the direction of motion. The gas is emitted at a constant mass ejection rate $\alpha$ and with constant speed $u$ relative to the trolley. The trolley and its supply of gas initially have a combined mass of $m_{0}$. How much time is spent ejecting gas before the trolley stops? [Assume that the trolley carries sufficient gas.]

Paper 4, Section II, 10C

comment(a) A mass $m$ is acted upon by a central force

$\mathbf{F}=-\frac{k m}{r^{3}} \mathbf{r}$

where $k$ is a positive constant and $\mathbf{r}$ is the displacement of the mass from the origin. Show that the angular momentum and energy of the mass are conserved.

(b) Working in plane polar coordinates $(r, \theta)$, or otherwise, show that the distance $r=|\mathbf{r}|$ between the mass and the origin obeys the following differential equation

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

where $h$ is the angular momentum per unit mass.

(c) A satellite is initially in a circular orbit of radius $r_{1}$ and experiences the force described above. At $\theta=0$ and time $t_{1}$, the satellite emits a short rocket burst putting it on an elliptical orbit with its closest distance to the centre $r_{1}$ and farthest distance $r_{2}$. When $\theta=\pi$ and the time is $t_{2}$, the satellite reaches the farthest distance and a second short rocket burst puts the rocket on a circular orbit of radius $r_{2}$. (See figure.) [Assume that the duration of the rocket bursts is negligible.]

(i) Show that the satellite's angular momentum per unit mass while in the elliptical orbit is

$h=\sqrt{\frac{C k r_{1} r_{2}}{r_{1}+r_{2}}}$

where $C$ is a number you should determine.

(ii) What is the change in speed as a result of the rocket burst at time $t_{1}$ ? And what is the change in speed at $t_{2}$ ?

(iii) Given that the elliptical orbit can be described by

$r=\frac{h^{2}}{k(1+e \cos \theta)}$

where $e$ is the eccentricity of the orbit, find $t_{2}-t_{1}$ in terms of $r_{1}, r_{2}$, and $k$. [Hint: The area of an ellipse is equal to $\pi a b$, where $a$ and b are its semi-major and semi-minor axes; these are related to the eccentricity by $\left.e=\sqrt{1-\frac{b^{2}}{a^{2}}} .\right]$

Paper 4, Section II, C

commentWrite down the expression for the momentum of a particle of rest mass $m$, moving with velocity $\mathbf{v}$ where $v=|\mathbf{v}|$ is near the speed of light $c$. Write down the corresponding 4-momentum.

Such a particle experiences a force $\mathbf{F}$. Why is the following expression for the particle's acceleration,

$\mathbf{a}=\frac{\mathbf{F}}{m}$

not generally correct? Show that the force can be written as follows

$\mathbf{F}=m \gamma\left(\frac{\gamma^{2}}{c^{2}}(\mathbf{v} \cdot \mathbf{a}) \mathbf{v}+\mathbf{a}\right)$

Invert this expression to find the particle's acceleration as the sum of two vectors, one parallel to $\mathbf{F}$ and one parallel to $\mathbf{v}$.

A particle with rest mass $m$ and charge $q$ is in the presence of a constant electric field $\mathbf{E}$ which exerts a force $\mathbf{F}=q \mathbf{E}$ on the particle. If the particle is at rest at $t=0$, its motion will be in the direction of $\mathbf{E}$ for $t>0$. Determine the particle's speed for $t>0$. How does the velocity behave as $t \rightarrow \infty$ ?

[Hint: You may find that trigonometric substitution is helpful in evaluating an integral.]

Paper 4, Section II, C

commentConsider an inertial frame of reference $S$ and a frame of reference $S^{\prime}$ which is rotating with constant angular velocity $\boldsymbol{\omega}$ relative to $S$. Assume that the two frames have a common origin $O$.

Let $\mathbf{A}$ be any vector. Explain why the derivative of $\mathbf{A}$ in frame $S$ is related to its derivative in $S^{\prime}$ by the following equation

$\left(\frac{d \mathbf{A}}{d t}\right)_{S}=\left(\frac{d \mathbf{A}}{d t}\right)_{S^{\prime}}+\omega \times \mathbf{A} .$

[Hint: It may be useful to use Cartesian basis vectors in both frames.]

Let $\mathbf{r}(t)$ be the position vector of a particle, measured from $O$. Derive the expression relating the particle's acceleration as observed in $S,\left(\frac{d^{2} \mathbf{r}}{d t^{2}}\right)_{S}$, to the acceleration observed in $S^{\prime},\left(\frac{d^{2} \mathbf{r}}{d t^{2}}\right)_{S^{\prime}}$, written in terms of $\mathbf{r}, \boldsymbol{\omega}$ and $\left(\frac{d \mathbf{r}}{d t}\right)_{S^{\prime}}$

A small bead of mass $m$ is threaded on a smooth, rigid, circular wire of radius $R$. At any given instant, the wire hangs in a vertical plane with respect to a downward gravitational acceleration $\mathbf{g}$. The wire is rotating with constant angular velocity $\boldsymbol{\omega}$ about its vertical diameter. Let $\theta(t)$ be the angle between the downward vertical and the radial line going from the centre of the hoop to the bead.

(i) Show that $\theta(t)$ satisfies the following equation of motion

$\ddot{\theta}=\left(\omega^{2} \cos \theta-\frac{g}{R}\right) \sin \theta .$

(ii) Find any equilibrium angles and determine their stability.

(iii) Find the force of the wire on the bead as a function of $\theta$ and $\dot{\theta}$.

Paper 4, Section II, C

commentA particle of mass $m$ follows a one-dimensional trajectory $x(t)$ in the presence of a variable force $F(x, t)$. Write down an expression for the work done by this force as the particle moves from $x\left(t_{a}\right)=a$ to $x\left(t_{b}\right)=b$. Assuming that this is the only force acting on the particle, show that the work done by the force is equal to the change in the particle's kinetic energy.

What does it mean if a force is said to be conservative?

A particle moves in a force field given by

$F(x)=\left\{\begin{array}{cc} -F_{0} e^{-x / \lambda} & x \geqslant 0 \\ F_{0} e^{x / \lambda} & x<0 \end{array}\right.$

where $F_{0}$ and $\lambda$ are positive constants. The particle starts at the origin $x=0$ with initial velocity $v_{0}>0$. Show that, as the particle's position increases from $x=0$ to larger $x>0$, the particle's velocity $v$ at position $x$ is given by

$v(x)=\sqrt{v_{0}^{2}+v_{e}^{2}\left(e^{-|x| / \lambda}-1\right)}$

where you should determine $v_{e}$. What determines whether the particle will escape to infinity or oscillate about the origin? Sketch $v(x)$ versus $x$ for each of these cases, carefully identifying any significant velocities or positions.

In the case of oscillatory motion, find the period of oscillation in terms of $v_{0}, v_{e}$, and $\lambda$. [Hint: You may use the fact that

$\int_{w}^{1} \frac{d u}{u \sqrt{u-w}}=\frac{2 \cos ^{-1} \sqrt{w}}{\sqrt{w}}$

for $0<w<1$.]

Paper 2, Section I, C

commentA particle $P$ with unit mass moves in a central potential $\Phi(r)=-k / r$ where $k>0$. Initially $P$ is a distance $R$ away from the origin moving with speed $u$ on a trajectory which, in the absence of any force, would be a straight line whose shortest distance from the origin is $b$. The shortest distance between $P$ 's actual trajectory and the origin is $p$, with $0<p<b$, at which point it is moving with speed $w$.

(i) Assuming $u^{2} \gg 2 k / R$, find $w^{2} / k$ in terms of $b$ and $p$.

(ii) Assuming $u^{2}<2 k / R$, find an expression for $P^{\prime}$ 's farthest distance from the origin $q$ in the form

$A q^{2}+B q+C=0$

where $A, B$, and $C$ depend only on $R, b, k$, and the angular momentum $L$.

[You do not need to prove that energy and angular momentum are conserved.]

Paper 2, Section II, C

comment(a) A moving particle with rest mass $M$ decays into two particles (photons) with zero rest mass. Derive an expression for $\sin \frac{\theta}{2}$, where $\theta$ is the angle between the spatial momenta of the final state particles, and show that it depends only on $M c^{2}$ and the energies of the massless particles. ( $c$ is the speed of light in vacuum.)

(b) A particle $P$ with rest mass $M$ decays into two particles: a particle $R$ with rest mass $0<m<M$ and another particle with zero rest mass. Using dimensional analysis explain why the speed $v$ of $R$ in the rest frame of $P$ can be expressed as

$v=c f(r), \quad \text { with } \quad r=\frac{m}{M}$

and $f$ a dimensionless function of $r$. Determine the function $f(r)$.

Choose coordinates in the rest frame of $P$ such that $R$ is emitted at $t=0$ from the origin in the $x$-direction. The particle $R$ decays after a time $\tau$, measured in its own rest frame. Determine the spacetime coordinates $(c t, x)$, in the rest frame of $P$, corresponding to this event.

Paper 2, Section II, C

commentAn axially symmetric pulley of mass $M$ rotates about a fixed, horizontal axis, say the $x$-axis. A string of fixed length and negligible mass connects two blocks with masses $m_{1}=M$ and $m_{2}=2 M$. The string is hung over the pulley, with one mass on each side. The tensions in the string due to masses $m_{1}$ and $m_{2}$ can respectively be labelled $T_{1}$ and $T_{2}$. The moment of inertia of the pulley is $I=q M a^{2}$, where $q$ is a number and $a$ is the radius of the

The motion of the pulley is opposed by a frictional torque of magnitude $\lambda M \omega$, where $\omega$ is the angular velocity of the pulley and $\lambda$ is a real positive constant. Obtain a first-order differential equation for $\omega$ and, from it, find $\omega(t)$ given that the system is released from rest.

The surface of the pulley is defined by revolving the function $b(x)$ about the $x$-axis, with

$b(x)= \begin{cases}a(1+|x|) & -1 \leqslant x \leqslant 1 \\ 0 & \text { otherwise. }\end{cases}$

Find a value for the constant $q$ given that the pulley has uniform mass density $\rho$.

Paper 4, Section I, A

commentGalileo releases a cannonball of mass $m$ from the top of the leaning tower of Pisa, a vertical height $h$ above the ground. Ignoring the rotation of the Earth but assuming that the cannonball experiences a quadratic drag force whose magnitude is $\gamma v^{2}$ (where $v$ is the speed of the cannonball), find the time for it to hit the ground in terms of $h, m, \gamma$ and $g$, the acceleration due to gravity. [You may assume that $g$ is constant.]

Paper 4, Section I, A

commentA rocket of mass $m(t)$ moving at speed $v(t)$ and ejecting fuel behind it at a constant speed $u$ relative to the rocket, is subject to an external force $F$. Considering a small time interval $\delta t$, derive the rocket equation

$m \frac{d v}{d t}+u \frac{d m}{d t}=F$

In deep space where $F=0$, how much faster does the rocket go if it burns half of its mass in fuel?

Paper 4, Section II, A

commentAn inertial frame $S$ and another reference frame $S^{\prime}$ have a common origin $O$, and $S^{\prime}$ rotates with angular velocity vector $\omega(t)$ with respect to $S$. Derive the results (a) and (b) below, where dot denotes a derivative with respect to time $t$ :

(a) The rates of change of an arbitrary vector $\mathbf{a}(t)$ in frames $S$ and $S^{\prime}$ are related by

$(\dot{\mathbf{a}})_{S}=(\dot{\mathbf{a}})_{S^{\prime}}+\boldsymbol{\omega} \times \mathbf{a} .$

(b) The accelerations in $S$ and $S^{\prime}$ are related by

$(\ddot{\mathbf{r}})_{S}=(\ddot{\mathbf{r}})_{S^{\prime}}+2 \boldsymbol{\omega} \times(\dot{\mathbf{r}})_{S^{\prime}}+(\dot{\boldsymbol{\omega}})_{S^{\prime}} \times \mathbf{r}+\boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{r}),$

where $\mathbf{r}(t)$ is the position vector relative to $O$.

Just after passing the South Pole, a ski-doo of mass $m$ is travelling on a constant longitude with speed $v$. Find the magnitude and direction of the sideways component of apparent force experienced by the ski-doo. [The sideways component is locally along the surface of the Earth and perpendicular to the motion of the ski-doo.]

Paper 4, Section II, A

comment(a) Writing a mass dimension as $M$, a time dimension as $T$, a length dimension as $L$ and a charge dimension as $Q$, write, using relations that you know, the dimensions of:

(i) force

(ii) electric field

(b) In the Large Hadron Collider at CERN, a proton of rest mass $m$ and charge $q>0$ is accelerated by a constant electric field $\mathbf{E} \neq \mathbf{0}$. At time $t=0$, the particle is at rest at the origin.

Writing the proton's position as $\mathbf{x}(t)$ and including relativistic effects, calculate $\dot{\mathbf{x}}(t)$. Use your answers to part (a) to check that the dimensions in your expression are correct.

Sketch a graph of $|\dot{\mathbf{x}}(t)|$ versus $t$, commenting on the $t \rightarrow \infty$ limit.

Calculate $|\mathbf{x}(t)|$ as an explicit function of $t$ and find the non-relativistic limit at small times $t$. What kind of motion is this?

(c) At a later time $t_{0}$, an observer in the laboratory frame sees a cosmic microwave photon of energy $E_{\gamma}$ hit the accelerated proton, leaving only a $\Delta^{+}$particle of mass $m_{\Delta}$ in the final state. In its rest frame, the $\Delta^{+}$takes a time $t_{\Delta}$ to decay. How long does it take to decay in the laboratory frame as a function of $q, \mathbf{E}, t_{0}, m, E_{\gamma}, m_{\Delta}, t_{\Delta}$ and $c$, the speed of light in a vacuum?

Paper 4, Section II, A

comment(a) A particle of mass $m$ and positive charge $q$ moves with velocity $\dot{\mathbf{x}}$ in a region in which the magnetic field $\mathbf{B}=(0,0, B)$ is constant and no other forces act, where $B>0$. Initially, the particle is at position $\mathbf{x}=(1,0,0)$ and $\dot{\mathbf{x}}=(0, v, v)$. Write the equation of motion of the particle and then solve it to find $\mathbf{x}$ as a function of time $t$. Sketch its path in $(x, y, z)$.

(b) For $B=0$, three point particles, each of charge $q$, are fixed at $(0, a / \sqrt{3}, 0)$, $(a / 2,-a /(2 \sqrt{3}), 0)$ and $(-a / 2,-a /(2 \sqrt{3}), 0)$, respectively. Another point particle of mass $m$ and charge $q$ is constrained to move in the $z=0$ plane and suffers Coulomb repulsion from each fixed charge. Neglecting any magnetic fields,

(i) Find the position of an equilibrium point.

(ii) By finding the form of the electric potential near this point, deduce that the equilibrium is stable.

(iii) Consider small displacements of the point particle from the equilibrium point. By resolving forces in the directions $(1,0,0)$ and $(0,1,0)$, show that the frequency of oscillation is

$\omega=A \frac{|q|}{\sqrt{m \epsilon_{0} a^{3}}},$

where $A$ is a constant which you should find.

[You may assume that if two identical charges $q$ are separated by a distance $d$ then the repulsive Coulomb force experienced by each of the charges is $q^{2} /\left(4 \pi \epsilon_{0} d^{2}\right)$, where $\epsilon_{0}$ is a constant.]

Paper 4, Section II, A

commentIn an alien invasion, a flying saucer hovers at a fixed point $S$, a height $l$ far above the White House, which is at point $W$. A wrecking ball of mass $m$ is attached to one end of a light inextensible rod, also of length $l$. The other end of the rod is attached to the flying saucer. The wrecking ball is initially at rest at point $B$, and the angle $W S B$ is $\theta_{0}$. At $W$, the acceleration due to gravity is $g$. Assume that the rotation of the Earth can be neglected and that the only force acting is Earth's gravity.

(a) Under the approximations that gravity acts everywhere parallel to the line $S W$ and that the acceleration due to Earth's gravity is constant throughout the space through which the wrecking ball is travelling, find the speed $v_{1}$ with which the wrecking ball hits the White House, in terms of the constants introduced above.

(b) Taking into account the fact that gravity is non-uniform and acts toward the centre of the Earth, find the speed $v_{2}$ with which the wrecking ball hits the White House in terms of the constants introduced above and $R$, where $R$ is the radius of the Earth, which you may assume is exactly spherical.

(c) Finally, show that

$v_{2}=v_{1}\left(1+\left(A+B \cos \theta_{0}\right) \frac{l}{R}+O\left(\frac{l^{2}}{R^{2}}\right)\right)$

where $A$ and $B$ are constants, which you should determine.

Paper 4, Section I, 3A

comment(a) Define an inertial frame.

(b) Specify three different types of Galilean transformation on inertial frames whose space coordinates are $\mathbf{x}$ and whose time coordinate is $t$.

(c) State the Principle of Galilean Relativity.

(d) Write down the equation of motion for a particle in one dimension $x$ in a potential $V(x)$. Prove that energy is conserved. A particle is at position $x_{0}$ at time $t_{0}$. Find an expression for time $t$ as a function of $x$ in terms of an integral involving $V$.

(e) Write down the $x$ values of any equilibria and state (without justification) whether they are stable or unstable for:

(i) $V(x)=\left(x^{2}-4\right)^{2}$

(ii) $V(x)=e^{-1 / x^{2}}$ for $x \neq 0$ and $V(0)=0$.

Paper 4, Section I, A

commentExplain what is meant by a central force acting on a particle moving in three dimensions.

Show that the angular momentum of a particle about the origin for a central force is conserved, and hence that its path lies in a plane.

Show that, in the approximation in which the Sun is regarded as fixed and only its gravitational field is considered, a straight line joining the Sun and an orbiting planet sweeps out equal areas in equal time (Kepler's second law).

Paper 4, Section II, A

commentThe position $\mathbf{x}=(x, y, z)$ and velocity $\dot{\mathbf{x}}$ of a particle of mass $m$ are measured in a frame which rotates at constant angular velocity $\omega$ with respect to an inertial frame. The particle is subject to a force $\mathbf{F}=-9 m|\boldsymbol{\omega}|^{2} \mathbf{x}$. What is the equation of motion of the particle?

Find the trajectory of the particle in the coordinates $(x, y, z)$ if $\boldsymbol{\omega}=(0,0, \omega)$ and at $t=0, \mathbf{x}=(1,0,0)$ and $\dot{\mathbf{x}}=(0,0,0)$.

Find the maximum value of the speed $|\dot{\mathbf{x}}|$ of the particle and the times at which it travels at this speed.

[Hint: You may find using the variable $\xi=x+i y$ helpful.]

Paper 4, Section II, A

commentWrite down the Lorentz force law for a charge $q$ travelling at velocity $\mathbf{v}$ in an electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$.

In a space station which is in an inertial frame, an experiment is performed in vacuo where a ball is released from rest a distance $h$ from a wall. The ball has charge $q>0$ and at time $t$, it is a distance $z(t)$ from the wall. A constant electric field of magnitude $E$ points toward the wall in a perpendicular direction, but there is no magnetic field. Find the speed of the ball on its first impact.

Every time the ball bounces, its speed is reduced by a factor $\gamma<1$. Show that the total distance travelled by the ball before it comes to rest is

$L=h \frac{q_{1}(\gamma)}{q_{2}(\gamma)}$

where $q_{1}$ and $q_{2}$ are quadratic functions which you should find explicitly.

A gas leak fills the apparatus with an atmosphere and the experiment is repeated. The ball now experiences an additional drag force $\mathbf{D}=-\alpha|\mathbf{v}| \mathbf{v}$, where $\mathbf{v}$ is the instantaneous velocity of the ball and $\alpha>0$. Solve the system before the first bounce, finding an explicit solution for the distance $z(t)$ between the ball and the wall as a function of time of the form

$z(t)=h-A f(B t)$

where $f$ is a function and $A$ and $B$ are dimensional constants, all of which you should find explicitly.

Paper 4, Section II, A

commentDefine the 4-momentum $P$ of a particle of rest mass $m$ and velocity $\mathbf{u}$. Calculate the power series expansion of the component $P^{0}$ for small $|\mathbf{u}| / c$ (where $c$ is the speed of light in vacuo) up to and including terms of order $|\mathbf{u}|^{4}$, and interpret the first two terms.

(a) At CERN, anti-protons are made by colliding a moving proton with another proton at rest in a fixed target. The collision in question produces three protons and an anti-proton. Assume that the rest mass of a proton is identical to the rest mass of an anti-proton. What is the smallest possible speed of the incoming proton (measured in the laboratory frame)?

(b) A moving particle of rest mass $M$ decays into $N$ particles with 4 -momenta $Q_{i}$, and rest masses $m_{i}$, where $i=1,2, \ldots, N$. Show that

$M=\frac{1}{c} \sqrt{\left(\sum_{i=1}^{N} Q_{i}\right) \cdot\left(\sum_{j=1}^{N} Q_{j}\right)}$

Thus, show that

$M \geqslant \sum_{i=1}^{N} m_{i}$

(c) A particle $A$ decays into particle $B$ and a massless particle 1 . Particle $B$ subsequently decays into particle $C$ and a massless particle 2 . Show that

$0 \leqslant\left(Q_{1}+Q_{2}\right) \cdot\left(Q_{1}+Q_{2}\right) \leqslant \frac{\left(m_{A}^{2}-m_{B}^{2}\right)\left(m_{B}^{2}-m_{C}^{2}\right) c^{2}}{m_{B}^{2}}$

where $Q_{1}$ and $Q_{2}$ are the 4-momenta of particles 1 and 2 respectively and $m_{A}, m_{B}, m_{C}$ are the masses of particles $A, B$ and $C$ respectively.

Paper 4, Section II, A

commentConsider a rigid body, whose shape and density distribution are rotationally symmetric about a horizontal axis. The body has mass $M$, radius $a$ and moment of inertia $I$ about its axis of rotational symmetry and is rolling down a non-slip slope inclined at an angle $\alpha$ to the horizontal. By considering its energy, calculate the acceleration of the disc down the slope in terms of the quantities introduced above and $g$, the acceleration due to gravity.

(a) A sphere with density proportional to $r^{c}$ (where $r$ is distance to the sphere's centre and $c$ is a positive constant) is launched up a non-slip slope of constant incline at the same time, level and speed as a vertical disc of constant density. Find $c$ such that the sphere and the disc return to their launch points at the same time.

(b) Two spherical glass marbles of equal radius are released from rest at time $t=0$ on an inclined non-slip slope of constant incline from the same level. The glass in each marble is of constant and equal density, but the second marble has two spherical air bubbles in it whose radii are half the radius of the marbles, initially centred directly above and below the marble's centre, respectively. Each bubble is centred half-way between the centre of the marble and its surface. At a later time $t$, find the ratio of the distance travelled by the first marble and the second. [ You may state without proof any theorems that you use and neglect the mass of air in the bubbles. ]

Paper 4, Section I, A

commentA tennis ball of mass $m$ is projected vertically upwards with initial speed $u_{0}$ and reaches its highest point at time $T$. In addition to uniform gravity, the ball experiences air resistance, which produces a frictional force of magnitude $\alpha v$, where $v$ is the ball's speed and $\alpha$ is a positive constant. Show by dimensional analysis that $T$ can be written in the form

$T=\frac{m}{\alpha} f(\lambda)$

for some function $f$ of a dimensionless quantity $\lambda$.

Use the equation of motion of the ball to find $f(\lambda)$.

Paper 4, Section I, A

commentConsider a system of particles with masses $m_{i}$ and position vectors $\mathbf{x}_{i}$. Write down the definition of the position of the centre of mass $\mathbf{R}$ of the system. Let $T$ be the total kinetic energy of the system. Show that

$T=\frac{1}{2} M \dot{\mathbf{R}} \cdot \dot{\mathbf{R}}+\frac{1}{2} \sum_{i} m_{i} \dot{\mathbf{y}}_{i} \cdot \dot{\mathbf{y}}_{i}$

where $M$ is the total mass and $\mathbf{y}_{i}$ is the position vector of particle $i$ with respect to $\mathbf{R}$.

The particles are connected to form a rigid body which rotates with angular speed $\omega$ about an axis $\mathbf{n}$ through $\mathbf{R}$, where $\mathbf{n} \cdot \mathbf{n}=1$. Show that

$T=\frac{1}{2} M \dot{\mathbf{R}} \cdot \dot{\mathbf{R}}+\frac{1}{2} I \omega^{2},$

where $I=\sum_{i} I_{i}$ and $I_{i}$ is the moment of inertia of particle $i$ about $\mathbf{n}$.

Paper 4, Section II, A

comment(a) A rocket moves in a straight line with speed $v(t)$ and is subject to no external forces. The rocket is composed of a body of mass $M$ and fuel of mass $m(t)$, which is burnt at constant rate $\alpha$ and the exhaust is ejected with constant speed $u$ relative to the rocket. Show that

$(M+m) \frac{d v}{d t}-\alpha u=0 .$

Show that the speed of the rocket when all its fuel is burnt is

$v_{0}+u \log \left(1+\frac{m_{0}}{M}\right)$

where $v_{0}$ and $m_{0}$ are the speed of the rocket and the mass of the fuel at $t=0$.

(b) A two-stage rocket moves in a straight line and is subject to no external forces. The rocket is initially at rest. The masses of the bodies of the two stages are $k M$ and $(1-k) M$, with $0 \leqslant k \leqslant 1$, and they initially carry masses $k m_{0}$ and $(1-k) m_{0}$ of fuel. Both stages burn fuel at a constant rate $\alpha$ when operating and the exhaust is ejected with constant speed $u$ relative to the rocket. The first stage operates first, until all its fuel is burnt. The body of the first stage is then detached with negligible force and the second stage ignites.

Find the speed of the second stage when all its fuel is burnt. For $0 \leqslant k<1$ compare it with the speed of the rocket in part (a) in the case $v_{0}=0$. Comment on the case $k=1$.

Paper 4, Section II, A

comment(a) Consider an inertial frame $S$, and a frame $S^{\prime}$ which rotates with constant angular velocity $\boldsymbol{\omega}$ relative to $S$. The two frames share a common origin. Identify each term in the equation

$\left(\frac{d^{2} \mathbf{r}}{d t^{2}}\right)_{S^{\prime}}=\left(\frac{d^{2} \mathbf{r}}{d t^{2}}\right)_{S}-2 \boldsymbol{\omega} \times\left(\frac{d \mathbf{r}}{d t}\right)_{S^{\prime}}-\boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{r})$

(b) A small bead $P$ of unit mass can slide without friction on a circular hoop of radius $a$. The hoop is horizontal and rotating with constant angular speed $\omega$ about a fixed vertical axis through a point $O$ on its circumference.

(i) Using Cartesian axes in the rotating frame $S^{\prime}$, with origin at $O$ and $x^{\prime}$-axis along the diameter of the hoop through $O$, write down the position vector of $P$ in terms of $a$ and the angle $\theta$ shown in the diagram $\left(-\frac{1}{2} \pi \leqslant \theta \leqslant \frac{1}{2} \pi\right)$.

(ii) Working again in the rotating frame, find, in terms of $a, \theta, \dot{\theta}$ and $\omega$, an expression for the horizontal component of the force exerted by the hoop on the bead.

(iii) For what value of $\theta$ is the bead in stable equilibrium? Find the frequency of small oscillations of the bead about that point.

Paper 4, Section II, A

commentA particle of unit mass moves under the influence of a central force. By considering the components of the acceleration in polar coordinates $(r, \theta)$ prove that the magnitude of the angular momentum is conserved. [You may use $\ddot{\mathbf{r}}=\left(\ddot{r}-r \dot{\theta}^{2}\right) \hat{\mathbf{r}}+(2 \dot{r} \dot{\theta}+r \ddot{\theta}) \hat{\boldsymbol{\theta}}$. ]

Now suppose that the central force is derived from the potential $k / r$, where $k$ is a constant.

(a) Show that the total energy of the particle can be written in the form

$E=\frac{1}{2} \dot{r}^{2}+V_{\mathrm{eff}}(r)$

Sketch $V_{\text {eff }}(r)$ in the cases $k>0$ and $k<0$.

(b) The particle is projected from a very large distance from the origin with speed $v$ and impact parameter $b$. [The impact parameter is the distance of closest approach to the origin in absence of any force.]

(i) In the case $k<0$, sketch the particle's trajectory and find the shortest distance $p$ between the particle and the origin, and the speed $u$ of the particle when $r=p$.

(ii) In the case $k>0$, sketch the particle's trajectory and find the corresponding shortest distance $\widetilde{p}$ between the particle and the origin, and the speed $\widetilde{u}$of the particle when $r=\widetilde{p}$.

(iii) Find $p \tilde{p}$ and $u \tilde{u}$ in terms of $b$ and $v$. [In answering part (iii) you should assume that $|k|$ is the same in parts (i) and (ii).]

Paper 4, Section II, A

comment(a) A photon with energy $E_{1}$ in the laboratory frame collides with an electron of rest mass $m$ that is initially at rest in the laboratory frame. As a result of the collision the photon is deflected through an angle $\theta$ as measured in the laboratory frame and its energy changes to $E_{2}$.

Derive an expression for $\frac{1}{E_{2}}-\frac{1}{E_{1}}$ in terms of $\theta, m$ and $c$.

(b) A deuterium atom with rest mass $m_{1}$ and energy $E_{1}$ in the laboratory frame collides with another deuterium atom that is initially at rest in the laboratory frame. The result of this collision is a proton of rest mass $m_{2}$ and energy $E_{2}$, and a tritium atom of rest mass $m_{3}$. Show that, if the proton is emitted perpendicular to the incoming trajectory of the deuterium atom as measured in the laboratory frame, then

$m_{3}^{2}=m_{2}^{2}+2\left(m_{1}+\frac{E_{1}}{c^{2}}\right)\left(m_{1}-\frac{E_{2}}{c^{2}}\right)$

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, C

commentWrite 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.

Paper 4, Section I, C

commentFind 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 $\frac{7}{10} M V^{2}$.

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

commentA 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

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

Paper 4, Section II, C

commentWrite 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 $x \approx \frac{1}{2} a t^{2}$.

Paper 4, Section II, C

commentConsider 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.

Paper 4, Section II, C

commentA 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 $\cdot \dot{\mathbf{u}} .]$

Paper 4, Section I, $3 C$

commentA particle of mass $m$ has charge $q$ and moves in a constant magnetic field B. Show that the particle's path describes a helix. In which direction is the axis of the helix, and what is the particle's rotational angular frequency about that axis?

Paper 4, Section I, $4 \mathrm{C}$

commentWhat is a 4-vector? Define the inner product of two 4-vectors and give the meanings of the terms timelike, null and spacelike. How do the four components of a 4-vector change under a Lorentz transformation of speed $v$ ? [Without loss of generality, you may take the velocity of the transformation to be along the positive $x$-axis.]

Show that a 4-vector that is timelike in one frame of reference is also timelike in a second frame of reference related by a Lorentz transformation. [Again, you may without loss of generality take the velocity of the transformation to be along the positive $x$-axis.]

Show that any null 4-vector may be written in the form $a(1, \hat{\mathbf{n}})$ where $a$ is real and $\hat{\mathbf{n}}$ is a unit 3-vector. Given any two null 4-vectors that are future-pointing, that is, which have positive time-components, show that their sum is either null or timelike.

Paper 4, Section II, C

commentDefine the 4-momentum of a particle and describe briefly the principle of conservation of 4-momentum.

A photon of angular frequency $\omega$ is absorbed by a particle of rest mass $m$ that is stationary in the laboratory frame of reference. The particle then splits into two equal particles, each of rest mass $\mathrm{\alpha m}$.

Find the maximum possible value of $\alpha$ as a function of $\mu=\hbar \omega / m c^{2}$. Verify that as $\mu \rightarrow 0$, this maximum value tends to $\frac{1}{2}$. For general $\mu$, show that when the maximum value of $\alpha$ is achieved, the resulting particles are each travelling at speed $c /\left(1+\mu^{-1}\right)$ in the laboratory frame.

Paper 4, Section II, C

commentA thin flat disc of radius $a$ has density (mass per unit area) $\rho(r, \theta)=\rho_{0}(a-r)$ where $(r, \theta)$ are plane polar coordinates on the disc and $\rho_{0}$ is a constant. The disc is free to rotate about a light, thin rod that is rigidly fixed in space, passing through the centre of the disc orthogonal to it. Find the moment of inertia of the disc about the rod.

The section of the disc lying in $r \geqslant \frac{1}{2} a,-\frac{\pi}{13} \leqslant \theta \leqslant \frac{\pi}{13}$ is cut out and removed. Starting from rest, a constant torque $\tau$ is applied to the remaining part of the disc until its angular speed about the axis reaches $\Omega$. Show that this takes a time

$\frac{3 \pi \rho_{0} a^{5} \Omega}{32 \tau}$

After this time, no further torque is applied and the partial disc continues to rotate at constant angular speed $\Omega$. Given that the total mass of the partial disc is $k \rho_{0} a^{3}$, where $k$ is a constant that you need not determine, find the position of the centre of mass, and hence its acceleration. From where does the force required to produce this acceleration arise?

Paper 4, Section II, C

commentA reference frame $S^{\prime}$ rotates with constant angular velocity $\boldsymbol{\omega}$ relative to an inertial frame $S$ that has the same origin as $S^{\prime}$. A particle of mass $m$ at position vector $\mathbf{x}$ is subject to a force $\mathbf{F}$. Derive the equation of motion for the particle in $S^{\prime}$.

A marble moves on a smooth plane which is inclined at an angle $\theta$ to the horizontal. The whole plane rotates at constant angular speed $\omega$ about a vertical axis through a point $O$ fixed in the plane. Coordinates $(\xi, \eta)$ are defined with respect to axes fixed in the plane: $O \xi$ horizontal and $O \eta$ up the line of greatest slope in the plane. Ensuring that you account for the normal reaction force, show that the motion of the marble obeys

$\begin{aligned} \ddot{\xi} &=\omega^{2} \xi+2 \omega \dot{\eta} \cos \theta, \\ \ddot{\eta} &=\omega^{2} \eta \cos ^{2} \theta-2 \omega \dot{\xi} \cos \theta-g \sin \theta \end{aligned}$

By considering the marble's kinetic energy as measured on the plane in the rotating frame, or otherwise, find a constant of the motion.

[You may assume that the marble never leaves the plane.]

Paper 4, Section II, C

commentA rocket of mass $m(t)$, which includes the mass of its fuel and everything on board, moves through free space in a straight line at speed $v(t)$. When its engines are operational, they burn fuel at a constant mass rate $\alpha$ and eject the waste gases behind the rocket at a constant speed $u$ relative to the rocket. Obtain the rocket equation

$m \frac{d v}{d t}-\alpha u=0$

The rocket is initially at rest in a cloud of space dust which is also at rest. The engines are started and, as the rocket travels through the cloud, it collects dust which it stores on board for research purposes. The mass of dust collected in a time $\delta t$ is given by $\beta \delta x$, where $\delta x$ is the distance travelled in that time and $\beta$ is a constant. Obtain the new equations

$\begin{aligned} \frac{d m}{d t} &=\beta v-\alpha \\ m \frac{d v}{d t} &=\alpha u-\beta v^{2} \end{aligned}$

By eliminating $t$, or otherwise, obtain the relationship

$m=\lambda m_{0} u \sqrt{\frac{(\lambda u-v)^{\lambda-1}}{(\lambda u+v)^{\lambda+1}}},$

where $m_{0}$ is the initial mass of the rocket and $\lambda=\sqrt{\alpha / \beta u}$.

If $\lambda>1$, show that the fuel will be exhausted before the speed of the rocket can reach $\lambda u$. Comment on the case when $\lambda<1$, giving a physical interpretation of your answer.

Paper 4 , Section II, B

comment(i) An inertial frame $S$ has orthonormal coordinate basis vectors $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$. A second frame $S^{\prime}$ rotates with angular velocity $\boldsymbol{\omega}$ relative to $S$ and has coordinate basis vectors $\mathbf{e}_{1}^{\prime}, \mathbf{e}_{2}^{\prime}, \mathbf{e}_{3}^{\prime}$. The motion of $S^{\prime}$ is characterised by the equations $d \mathbf{e}_{i}^{\prime} / d t=\boldsymbol{\omega} \times \mathbf{e}_{i}^{\prime}$ and at $t=0$ the two coordinate frames coincide.

If a particle $P$ has position vector $\mathbf{r}$ show that $\mathbf{v}=\mathbf{v}^{\prime}+\boldsymbol{\omega} \times \mathbf{r}$ where $\mathbf{v}$ and $\mathbf{v}^{\prime}$ are the velocity vectors of $P$ as seen by observers fixed respectively in $S$ and $S^{\prime}$.

(ii) For the remainder of this question you may assume that $\mathbf{a}=\mathbf{a}^{\prime}+2 \boldsymbol{\omega} \times \mathbf{v}^{\prime}+\boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{r})$ where $\mathbf{a}$ and $\mathbf{a}^{\prime}$ are the acceleration vectors of $P$ as seen by observers fixed respectively in $S$ and $S^{\prime}$, and that $\omega$ is constant.

Consider again the frames $S$ and $S^{\prime}$ in (i). Suppose that $\omega=\omega \mathbf{e}_{3}$ with $\omega$ constant. A particle of mass $m$ moves under a force $\mathbf{F}=-4 m \omega^{2} \mathbf{r}$. When viewed in $S^{\prime}$ its position and velocity at time $t=0$ are $\left(x^{\prime}, y^{\prime}, z^{\prime}\right)=(1,0,0)$ and $\left(\dot{x}^{\prime}, \dot{y}^{\prime}, \dot{z}^{\prime}\right)=(0,0,0)$. Find the motion of the particle in the coordinates of $S^{\prime}$. Show that for an observer fixed in $S^{\prime}$, the particle achieves its maximum speed at time $t=\pi /(4 \omega)$ and determine that speed. [Hint: you may find it useful to consider the combination $\zeta=x^{\prime}+i y^{\prime}$.]

Paper 4, Section $I$, B

commentA frame $S^{\prime}$ moves with constant velocity $v$ along the $x$ axis of an inertial frame $S$ of Minkowski space. A particle $P$ moves with constant velocity $u^{\prime}$ along the $x^{\prime}$ axis of $S^{\prime}$. Find the velocity $u$ of $P$ in $S$.

The rapidity $\varphi$ of any velocity $w$ is defined by $\tanh \varphi=w / c$. Find a relation between the rapidities of $u, u^{\prime}$ and $v$.

Suppose now that $P$ is initially at rest in $S$ and is subsequently given $n$ successive velocity increments of $c / 2$ (each delivered in the instantaneous rest frame of the particle). Show that the resulting velocity of $P$ in $S$ is

$c\left(\frac{e^{2 n \alpha}-1}{e^{2 n \alpha}+1}\right)$

where $\tanh \alpha=1 / 2$.

[You may use without proof the addition formulae $\sinh (a+b)=\sinh a \cosh b+\cosh a \sinh b$ and $\cosh (a+b)=\cosh a \cosh b+\sinh a \sinh b$.]

Paper 4, Section I, B

commentA hot air balloon of mass $M$ is equipped with a bag of sand of mass $m=m(t)$ which decreases in time as the sand is gradually released. In addition to gravity the balloon experiences a constant upwards buoyancy force $T$ and we neglect air resistance effects. Show that if $v(t)$ is the upward speed of the balloon then

$(M+m) \frac{d v}{d t}=T-(M+m) g .$

Initially at $t=0$ the mass of sand is $m(0)=m_{0}$ and the balloon is at rest in equilibrium. Subsequently the sand is released at a constant rate and is depleted in a time $t_{0}$. Show that the speed of the balloon at time $t_{0}$ is

$g t_{0}\left(\left(1+\frac{M}{m_{0}}\right) \ln \left(1+\frac{m_{0}}{M}\right)-1\right)$

[You may use without proof the indefinite integral $\int t /(A-t) d t=-t-A \ln |A-t|+C .$ ]

Paper 4, Section II, B

comment(a) Let $S$ with coordinates $(c t, x, y)$ and $S^{\prime}$ with coordinates $\left(c t^{\prime}, x^{\prime}, y^{\prime}\right)$ be inertial frames in Minkowski space with two spatial dimensions. $S^{\prime}$ moves with velocity $v$ along the $x$-axis of $S$ and they are related by the standard Lorentz transformation:

$\left(\begin{array}{c} c t \\ x \\ y \end{array}\right)=\left(\begin{array}{ccc} \gamma & \gamma v / c & 0 \\ \gamma v / c & \gamma & 0 \\ 0 & 0 & 1 \end{array}\right)\left(\begin{array}{c} c t^{\prime} \\ x^{\prime} \\ y^{\prime} \end{array}\right), \quad \text { where } \gamma=\frac{1}{\sqrt{1-v^{2} / c^{2}}} .$

A photon is emitted at the spacetime origin. In $S^{\prime}$ it has frequency $\nu^{\prime}$ and propagates at angle $\theta^{\prime}$ to the $x^{\prime}$-axis.

Write down the 4 -momentum of the photon in the frame $S^{\prime}$.

Hence or otherwise find the frequency of the photon as seen in $S$. Show that it propagates at angle $\theta$ to the $x$-axis in $S$, where

$\tan \theta=\frac{\tan \theta^{\prime}}{\gamma\left(1+\frac{v}{c} \sec \theta^{\prime}\right)}$

A light source in $S^{\prime}$ emits photons uniformly in all directions in the $x^{\prime} y^{\prime}$-plane. Show that for large $v$, in $S$ half of the light is concentrated into a narrow cone whose semi-angle $\alpha$ is given by $\cos \alpha=v / c$.

(b) The centre-of-mass frame for a system of relativistic particles in Minkowski space is the frame in which the total relativistic 3-momentum is zero.

Two particles $A_{1}$ and $A_{2}$ of rest masses $m_{1}$ and $m_{2}$ move collinearly with uniform velocities $u_{1}$ and $u_{2}$ respectively, along the $x$-axis of a frame $S$. They collide, coalescing to form a single particle $A_{3}$.

Determine the velocity of the centre-of-mass frame of the system comprising $A_{1}$ and $A_{2}$.

Find the speed of $A_{3}$ in $S$ and show that its rest mass $m_{3}$ is given by

$m_{3}^{2}=m_{1}^{2}+m_{2}^{2}+2 m_{1} m_{2} \gamma_{1} \gamma_{2}\left(1-\frac{u_{1} u_{2}}{c^{2}}\right),$

where $\gamma_{i}=\left(1-u_{i}^{2} / c^{2}\right)^{-1 / 2}$

Paper 4, Section II, B

comment(a) A rigid body $Q$ is made up of $N$ particles of masses $m_{i}$ at positions $\mathbf{r}_{i}(t)$. Let $\mathbf{R}(t)$ denote the position of its centre of mass. Show that the total kinetic energy of $Q$ may be decomposed into $T_{1}$, the kinetic energy of the centre of mass, plus a term $T_{2}$ representing the kinetic energy about the centre of mass.

Suppose now that $Q$ is rotating with angular velocity $\boldsymbol{\omega}$ about its centre of mass. Define the moment of inertia $I$ of $Q$ (about the axis defined by $\boldsymbol{\omega}$ ) and derive an expression for $T_{2}$ in terms of $I$ and $\omega=|\omega|$.

(b) Consider a uniform rod $A B$ of length $2 l$ and mass $M$. Two such rods $A B$ and $B C$ are freely hinged together at $B$. The end $A$ is attached to a fixed point $O$ on a perfectly smooth horizontal floor and $A B$ is able to rotate freely about $O$. The rods are initially at rest, lying in a vertical plane with $C$ resting on the floor and each rod making angle $\alpha$ with the horizontal. The rods subsequently move under gravity in their vertical plane.

Find an expression for the angular velocity of rod $A B$ when it makes angle $\theta$ with the floor. Determine the speed at which the hinge strikes the floor.

Paper 4, Section II, B

(a) A particle $P$ of unit mass moves in a plane with polar coordinates $(r, \theta)$. You may assume that the radial and angular components of the acceleration are given by $\left(\ddot{r}-r \dot{\theta}^{2}, r \ddot{\theta}+2 \dot{r} \dot{\theta}\right)$, where the dot denotes $d / d t$. The particle experiences a central force corresponding to a potential $V=V(r)$.

(i) Prove that $l=r^{2} \dot{\theta}$ is constant in time and show that the time dependence of the radial coordinate $r(t)$