PAIN poetry

2006: A Physics Curriculum

We begin by revising the notion of force,
And Newtonian concepts of motion, of course.
Basic equations will never be spared,
Like ess equals yoo tee plus half ay tee squared.

\[s = ut + \frac{1}{2}at^2\]

Then things get better, I’m sure you’ll agree.
Let kinetic be T and potential be V.
Take the difference of these two and call this guy L.
At the coming equation you’ll want to excel.

Write L as a function of q and q-dot.
Find dee L dee q-dot, and then once this is got,
Dee dee tee of this guy, you will see that it’s true,
Is simply the same as dee L by dee q!

\[L=T-V=L(q,\dot{q})\]

\[\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) =\frac{\partial L}{\partial q}\]

Then thermodynamics, the ideal gas,
You’ll be given the pressure and particle’s mass,
And density, volume, and then asked to show
If the temperature therefore is high or is low.

\[\rho = m/V, \quad PV = nKT\]

If the system is large, you will need to find Z,
That function from which many things can be read.
Z is the sum of all terms— Can you guess?
Ee to the negative beta ee ess!

\[Z = \sum_s \exp(-\beta E_s)\]

Next about optics you’ll learn of this fact:
When the medium changes, a wave will refract.
The law that describes this is named after Snell,
And it’s simple and short but it works very well.

Take n in part 1 over n in part 2,
And look how the angle got smaller or grew.
All students will know then, yes even the worst,
That it’s sine of the second on sine of the first!

\[\frac{n_1}{n_2} = \frac{\sin(\theta_2)}{\sin(\theta_1)}\]

From down in the doldrums did physics revive
With what Einstein published in nineteen oh five.
Light in a vacuum goes at the same speed,
Regardless of reference, religion or creed.

"It follows," said Albert (and now we can test),
"That time will go fastest when you are at rest."
A moving clock slows by a factor that he shared
Of one on root one minus v squared on c squared.

\[t' = \frac{t}{\sqrt{1 - v^2/c^2}}\]

There’s a lot more to physics than just all of that,
Like atoms and photons and Schrödinger’s cat.
But it takes me forever to get this to rhyme,
And to go any further I’d need much more time.

2007: The Adventures of John the Electron

In a lab in amongst all the dials and the clocks
And the laser beams sat a mysterious box.
From here it just looks like a regular cube,
But inside you will see it’s a cathode ray tube.

In the CRT’s wire a current was flowing
And John the Electron’s excitement was growing.
The electrons all played with the force that they shared
(k times q-one times q-two on r squared),
But all of them now only thought of their dream:
Leaving the wire and joining the beam.

\[F = k\frac{q_1 q_2}{r^2}\]

You may think this life is a little bit vapid;
For them it's like paddling a white-water rapid!
John bounced off some charges and then he was free!
He yelled out in triumph and chortled with glee!
He zoomed through the vacuum and moved very fast,
Enjoying the moment; forgetting the past.

But ahead was a hurdle for this little caper:
A magnetic field pointed into the paper.
“Oh no!” said poor John. “It's an end to my fun!
My free particle days are all dusted and done!
Look at my path and the field, if you please,
The angle between us is ninety degrees!
Then” (you should know) “an electromag quirk'll
Consign me to move in a boring old circle.
I know I can't always have freedom, of course,
But this'll become a centripetal force!”

The circular fate of poor Johnny was sealed
By his charge times his speed times the magnetic field.
If you make this the same as the force that you need
(That is m over r, times the square of the speed),
You can find what the size of John's circle must be:
It's one over q, over B, times mv.

\[F = qvB = \frac{m}{r}v^2\]

\[\implies r = \frac{mv}{qB}\]

It looked like John wouldn't be able to cope,
But still he clung on to one last final hope.
“If the magnetic region does not get much bigger,
It's easily seen (and with excellent rigour)
My trajectory's motion will not become bound,
For all it will do is just turn me around.”

John entered the field and he swung to the right,
Hoping and hoping with all of his might
That approaching the end of his half-revolution
He'd sense in the field any sharp diminution.

But a new pair of magnets was placed by the first,
Realising John's fears, the worst of the worst!
Any small fraction of lingering doubt
Was destroyed when the CRT's voltage went out.
The lives of electrons come with this great risk:
Tracing out a trajectory shaped like a disc.

The story to this point might look nice and neat,
But a rotating charge has to radiate heat.
Abraham published (along with Lorentz)
A formula making a good deal of sense.
I think that all people should find it concerning
That students will skip this – a gap in their learning.
One sixth a-dot times q squared is what should be taught,
Times one on pi c cubed times epsilon-nought.

\[F_{\mathrm{rad}} = \frac{q^2 \dot{a}}{6\pi \varepsilon_0 c^3}\]

The upshot of this was a deceleration
(Though nothing did change to the time of rotation.
Now don't look at me to correct and condemn,
The angular frequency's qB on m.)

\[\omega = \frac{qB}{m}\]

The spiral got small and though John did protest,
He kept slowing down and he finished at rest.

2008: Spotter the Otter

By a river that flowed in old Ottoman land,
A big group of otters relaxed on the sand.
The otter called Fay and the otter called Caleb,
The otter called Delf and the otter called Waleb,
The otter called Vlistance, the otter called Froo,
The otter called Marcus Aurelius too,
All of them turned to the otter called Spotter,
The otter who wasn't your ordinary otter.
Spotter the Otter had mastered the stars,
And the comets and Saturn and Venus and Mars.

The otters were gathered this fine and clear night
To learn about all of the stars in their sight.
“Please Spotter, explain,” said the otter called Vlistance,
“How stars that we see here came into existence.”

And Spotter the Otter then pointed his 'scope,
And all of the otters were lifted with hope,
'Cause they loved to zoom in on the sky in the night.
One of them looked and then squeaked with delight.

Spotter let everyone look and then said,
“What you see there is the famous Horsehead!

“A nebula has a collection of dust,
But it’s all very cold and it doesn’t combust.
If there’s enough dust, then maybe, perhaps,
Gravity makes it all start to collapse.”

“But Spotter,” asked one little otter called Fay,
“It can’t just be random — there must be a way
How to tell if the cloud will fall in or stay static.
The workings of physics must not be erratic.”

“You’re entirely correct there,” said Spotter to Fay,
“And now I’ll explain that with no more delay.
The critical mass was discovered by Jeans,
But it might be too hard now to see what it means.
The formula’s plainly a long one,” said he.
“You start first by taking the square root of 3,”
And continuing with a peculiar intensity,
“Divided by 4 timesed by pi times the density,
Then five times the constant of Boltzmann times T,
Divided by m timesed by Newton's big G,
Raised to the power of three over two,
You've now got the Jeans mass, I swear that it's true!”

\[M_J = \sqrt{\frac{3}{4\pi \rho}} \left( \frac{5kT}{Gm}\right)^{3/2}\]

“But what happens next,” asked the otter called Delf,
“When the gaseous cloud's fallen in on itself?”

“You're going too fast now,” was Spotter's reply,
And Spotter the Otter then gave a small sigh.
“The cloud starts to collapse, and some sub-clouds of gas,
Sufficiently denser and with enough mass,
All clump together and start to rotate.
It makes the collapsing then start to abate:
It spins and it spins and gets hotter and hotter,
Until it's more stable,” said Spotter the Otter.
"So the cloud as a whole will keep up its collapse,
But there might be some protostars, maybe, perhaps.

“The protostar keeps on attracting more mass
From all the surrounding molecular gas.
It slowly contracts and so generates heat,
Until this small part of its life is complete.
Its time as a protostar’s near its conclusion:
Deuterium then starts its nuclear fusion.
The accretion keeps making it more and more massive,
The internal behaviour gets less and less passive.
The hydrogen then starts to fuse and we see
A star that is clear to you and to me!”

The otters' instruction had come to an end;
They were happy with what they could now comprehend.
So the otter called Bob and the otter called Pooley,
The otter called Squinky-Di-Alla Mavooley,
The otter called Plok and the otter called Totter,
All knew a lot more thanks to Spotter the Otter.

2009: Statistical Mechanics

The United Italian Republic of Bosporus
Had always blue skies, and always was prosperous.
The assortment of people there bordered on mystical;
Everyone there liked their physics statistical.

And if you were there, you could wander around
And listen and hear all the noises and sound,
And see Benjamin Harrison looking at wires,
Checking connections to stop any fires.
“I'm always at power lines 'cause I'm a faultsman;
I'm also an expert on Maxwell and Boltzmann,”
Said Benjamin just before wiping his brow.
“Consider some gaseous particles now.
Take all the velocities, x, y, and z.
There's too many of them to keep in your head.
But let each of them follow a Gaussian law,
And consider the whole – we know more than before.
The probabilistic resulting relation
Is (up to an overall normalisation)
The Gaussian term that you’d look for, indeed,
But also, before it, the square of the speed.
The hotter the gas is, the greater the spread
Of the speeds of the particles,” Benjamin said.

\[f(v) = \sqrt{\frac{2}{\pi}} \left( \frac{m}{kT}\right)^{3/2} v^2 \exp(-mv^2/kT)\]

But all of that's only Newtonian stuff,
And some of the people considered it fluff,
With all of its classical characteristics.
Those people also knew quantum statistics.

One of these people was dressed in a gown:
England’s first Queen Elizabeth, wearing a crown.
But you’d not try the Palace attempting to reach her –
You’d go to the school where she works as a teacher.
She’d sit near a blackboard and talk with a grin
On a theorem relating statistics to spin.
“Let us imagine some fermions here.
Let’s both have one each!” she said with good cheer.
“If you don’t like your one and swap it for mine,
The overall state gains a negative sign.
So identical states with a spin of a half
Must have already vanished!” she said with a laugh.
“They’re not at all normal,” she said, leaning back,
“And obey the relation of Fermi-Dirac.
If the particles go in states low up to high,
We can’t say exactly the number in i,
But the average is one on one added to e
To the epsilon-i minus mu on kT.”

\[N_i = \frac{1}{1 + \exp\left(\frac{\varepsilon_i - \mu}{kT}\right)}\]

“That’s not all of quantum mechanics,” said Thomas.
“There’s also statistics of Bose to learn from us.”
Saint Thomas Aquinas’s studies scholastic
Had taken a turn for the not-so-monastic.
“If there’s one boson here and another one there,
The system’s the same on exchanging the pair.
The problem of energies has a solution:
The particles follow a Bose distribution.
Take Fermi-Dirac,” said Saint Thomas Aquinas.
“What once was a plus has turned into a minus!

\[N_i = \frac{1}{-1 + \exp\left(\frac{\varepsilon_i - \mu}{kT}\right)}\]

“Ultra-cold temperatures start the creation
Of particles driven to Bose condensation.
The critical temperature’s straight out of Heaven.
The constant has digits of 5, 2, and 7.”
Then he continued, a glint in his eye.
“Take Planck’s constant squared over m times two pi
(And here it is written without any surds)
Timesed by the density raised to two thirds!”

\[kT_C = 0.527 \frac{\hbar^2}{2\pi m} \left(\frac{N}{V}\right)^{2/3} \]

Despite the appearance you might have inferred,
It’s not at all true that the workings you’ve heard
Of the Queen and the Saint and the old electrician
Define on all stat mech a simple partition.

To learn about all of this further diversity,
Study some stat mech when at university.

2010: Young’s Double Slit

Near the middle of London in eighteen-oh-one,
An experiment studying light had begun.
Before then the model most thought to be right
Was Newton’s corpuscular theory of light.
Newton had seemingly put to the grave
The notion from Huygens that light is a wave.
But one Thomas Young caused a massive retraction,
With light rays that underwent two-slit diffraction.

When waves pass through slits that are suitably narrow,
Their motion’s no longer as straight as an arrow.
Some minor diffractive effects notwithstanding,
They move out in arcs that all keep on expanding.
In a double-slit system, from each little gap,
The arcs all expand, so they soon overlap.
And when waves come together, they must interfere –
They might both add up or they might disappear.

It’s not hard to determine (assuming coherence)
The pattern that’s made by the waves’ interference.
If the waves can all hit and then light up a screen,
A series of light and dark fringes is seen.

In Thomas Young’s lab back in eighteen-oh-one,
Young tried this diffraction with light from the sun.
He saw all the fringes, and this observation
Persuaded him that the correct explanation
Of light is it’s simply, entirely composed
Of waves, not the particles Newton supposed.
None could then argue: the pattern appears,
And the wave theory ruled for some one hundred years.

But the theory that light is just wave-like was wrecked
When we first saw the photoelectric effect.
As well as being wave-like, it’s thought to be true
Now that light comes in small massless particles too.
It was therefore conjectured that matter could act
As a wave and might therefore refract or diffract.
And indeed the diffraction works not just with photons
But also electrons (or neutrons or protons).

And making this further bizarrely sublime,
It works if they only go one at a time!
Each single electron just makes a small dot,
But after a while you’ll end up with a lot.
And if you can wait with enough perseverance,
The dots will add up to show wave interference –
With single electrons we see the same sight
Of the fringes we’d normally make using light.

This should befuddle you out of your wits,
For it means the electrons all passed through both slits.
It’s not just not knowing which one they go through –
To self-interfere they must go through the two.
A particle in such an oddball condition
Is said to exist in a superposition.

You might think a detector could give us a clue,
And tell us the slit each one really went through.
But such an addition would come at a price:
The forced interaction with such a device
Will make the electron state start decohering,
And quickly prevent it from self-interfering.
The screen now will only show two brighter bits –
One corresponding to each of the slits.

Whether or not we can get all the fringes
Displayed on the screen is a question that hinges
On letting the “particle” act as a wave.
If we find its location, we make it behave
As a classical particle, thereby ensuring
That all the results will be normal and boring.

These experiments all with particular clarity
Show us the concept of complementarity.
We can only see one of the path information
And wave-like results in the fringes’ creation:
We’re given this choice and it’s sad but it’s true
That the universe doesn’t allow us the two.

So from Young’s demonstration in eighteen-oh-one,
To all of the later experiments done,
There’s nothing so constant and no bigger hit
In laboratory physics than Young’s double-slit.

2011: The Zhukovsky Transform

There's a great lot of detail in rocket design,
But tonight, for our purposes, it'll be fine
To consider a model discovered by Moore
(Who thought not about space but of rockets in war).

Assuming the mass of a rocket decreases
Consistently during the time it releases
Exhaust that's created from burning its fuel,
Conserving momentum then gives us a rule:

Δv = v_e * ln(m₀ / m₁)

The change in the rocket speed, called delta-v,
Will be the effective exhaust speed v_e
Times the natural log of m-nought on m-one,
That's the mass at the start on the mass when it's done.

The log in that formula's quite consequential:
The growth in the fuel that you need's exponential
For larger desired delta-v's.  And in spite
Of its seeming quite similar, orbital flight
Requires much greater speeds than the ones that were true
Of the spaceflight achieved by the German V-2.

The V-2 was a single-stage rocket designed
By one Wernher von Braun. They inflicted a kind
Of emotional fear in the Second World War
As they headed to London at speeds of Mach 4.

These rockets were fueled by a mixture of three
Quarters ethanol, one quarter water, to be
Quickly burnt with near five tonnes of LOX.  At their high
Point V-2's would be moving in freefall and fly
Almost ninety kilometres up.

                              Once the war
Had been ended, America tried to ensure
That the brightest of Germany's scientists went
To the West, which would crucially also prevent
Them from working for Russia. Recruits from this mission
Included von Braun, who was given permission
To join the American Army in spite
Of his work with the Nazis, a sizeable blight
That was wiped from his record.

                                 In American hands,
The V-2's and equipment were moved to White Sands
In New Mexico, ready for firing in June
'46. With their vertical launches, they soon
Had the first ever spaceflight photography,
Followed soon after by UV spectrography.

But for orbital flight, the V-2's were too slow,
As some simple Newtonian physics will show.

GM/r² = v²/r

Since GM on r-squared must be v-squared on r,
Then as long as we have an idea of how far
Above Earth it will finish, it's easy to see
What the necessary speed while in orbit will be.
If you plug in the numbers, you'll see it works out
That for typical altitudes, speeds of about
Eight thousand metres per second are needed --
That's several times more than V-2's had succeeded
In reaching.

              That log you'll remember implies
A great need for fuel.  So it's not a surprise
That it wasn't till almost a dozen years later
That Sputnik was launched.  It was one of the greater
Successes the Soviets had; consternation
In Washington soon led to NASA's creation.

The satellite Sputnik itself was quite small:
Less than two feet across, in the shape of a ball.
Attached were a pair of antennas which each
Had two parts, and were set so that signals would reach
The Earth's surface regardless of orientation.
The rocket which launched it, a modification
Of Russia's first ICBM, employed four
Strap-on boosters for launch, which together held more
Than forty-two tonnes of its fuel, a refined
Kerosene with no linear alkanes, designed
To be burnt in exactly two minutes with two
Times its volume in LOX.  While the second stage drew
On the same two propellants, the rocket this stage
Would burn for five minutes and then disengage.

The orbit of Sputnik began to decay
A little bit into its ninety-third day.
Its orbital period measured a touch
Over 96 minutes, and therefore as such,
It lasted a thousand four hundred complete
Revolutions, burned up, and became obsolete.

When the cosmonaut Yuri Gagarin became
The first man in space, one could credibly claim
That the Russians had further established their power
In the Space Race.  In circling the world in an hour
And a half, the Lieutenant Gagarin compelled
The Americans into response. It propelled
Their space programme: Kennedy started a plan
That would see the Americans "landing a man
On the moon and returning him safely to ... Earth."
With a president thinking the programme was worth
The expenditure, all of the years that would follow
Were hectic for people who worked on Apollo.

The rest of the decade was not all smooth sailing.
The worst was the crew of Apollo 1 failing
To put out a fire in the cabin which led
To the astronauts trapped in there ending up dead.

Remarkably, though, the direction of Wernher
Von Braun kept Apollo on schedule, the sterner
Procedures on safety not stopping his staunch
Commitment to seeing the moon mission launch
By the end of the 60's.  It came to fruition
On 16 July '69.  The ignition
Occurred just a couple of seconds before
The clock ticked to nine thirty-two.  Then the roar
Of the Saturn V rocket was heard as it burned
Its refined kerosene with its LOX.

                                    As we learned
From the rocket equation, the fuel that is needed
Scales with the payload; Apollo's exceeded
All previous payloads.  All up more than two
Thousand tonnes of propellant were used as it flew
To sixty kilometres altitude.  Nearly
Some five hundred more got the spacecraft to merely
An orbit round Earth.  

                       Then it came time to start
The Trans-Lunar Injection, which made it depart
The Earth's orbit and head to the moon.  By the end
Of this burn, the spacecraft would have to contend
For propulsion with rockets on board
The command/service module.

                             Despite being ignored
Early on, the designers all chose the outstanding
Idea of employing a module for landing,
Which saw the command/service module remaining
In orbit. The Eagle detached, while containing
The astronauts Armstrong and Aldrin, descended,
Then after some moonwalks and sleep time, ascended
Back up and re-docked with the orbiter.  Choosing
This setup perhaps might have saved them from losing
The Space Race -- the fuel savings meant that the mission
Fulfilled the late Kennedy's early ambition.

No longer useful, the Eagle was sent
To an orbit they didn't keep track of. It spent
Its last days by decaying in orbit until,
In an unknown location, it crashed, and lies still.

The last major burn was the Trans-Earth Injection,
Followed by only the odd course correction,
The astronauts finishing this most terrific
Of stories by splashing down in the Pacific. 

Subsequent spaceflight just doesn't compare
To Apollo.  The missions, though, didn't end there --
Voyager, Skylab, the shuttles, and Mir --
But to rhyme all of that?  Well I'll wait till next year.