For your viewing pleasure, and perhaps for your educational pleasure, I've included some of my physics limericks here. Some are funny. Some are stupid. But at least they are all physically accurate (give or take). Many of them can be found scattered throughout the textbook I wrote for the Physics-16 course here at Harvard. (Some more physics humor is located here.)
No pain, no gain. Alas, there are no shortcuts to learning physics...
The ad said, for one little fee,
You can skip all that course-work ennui.
So send your tuition,
For boundless fruition!
Get your mail-order physics degree!
Always check your units!
Your units are wrong! cried the teacher.
Your church weighs six joules — what a feature!
And the people inside
Are four hours wide,
And eight gauss away from the preacher!
And check the limiting cases, too!
The lemmings get set for their race.
With one step and two steps they pace.
They take three and four,
And then head on for more,
Without checking the limiting case.
When converting the units of a quantity, all you have to do is multiply by 1 in an appropriately chosen form. For example, in converting minutes to seconds, just multiply 1min by 60s/1min (which equals 1), to obtain 1min=(1min)(60s/1min)=60s. The minutes cancel, and you're left with seconds. Of course, you could also multiply by 1 in the form of (1min/60s) to obtain 1min=(1min)(1min/60s)=1min2/60s. This does indeed equal 1 minute, but no one would know what you were talking about if you said you could run a 1min2/10s mile.
To figure the inches you’ve run,
Or to find the slug mass of the sun,
Forget your aversion
To unit conversion.
Just multiply (wisely!) by 1.
People tend to rely a bit too much on computers and calculators nowadays, without pausing to think about what is actually going on in a problem.
The skill to do math on a page
Has declined to the point of outrage.
Are solved on Math'matica,
And on birthdays we don't know our age.
Enrico Fermi was known for his ability to estimate things quickly and produce an order-of-magnitude guess with only minimal calculation. Hence, a problem where the goal is to simply obtain the nearest power-of-10 estimate is known as a "Fermi problem". Of course, sometimes in life you need to know things to better accuracy than the nearest power of 10...
How Fermi could estimate things!
Like the well-known Olympic ten rings,
And the one-hundred states,
And weeks with ten dates,
And birds that all fly with one... wings.
Newton's first law, "A body moves with constant velocity (which may be zero) unless acted on by a force," appears at first glance to be rather vacuous. In the words of Sir Arthur Stanley Eddington, it says that "every particle continues in its state of rest or uniform motion in a straight line except insofar that it doesn't." But the first law actually does have some content. In particular, it gives a definition of an "inertial frame" (which is defined to be one in which the first law holds). This inertial frame is then used as the setting for Newton's second law.
For things moving free or at rest,
Observe what the first law does best.
It defines a key frame,
Inertial by name,
Where the second law then is expressed.
Atwood's machines can be pretty hairy. But no matter how complicated they get, there are only two things you need to do to solve them: (1) Write down the F=ma equations for all the masses (which may involve relating the tensions in various strings), and (2) relate the accelerations of the masses, using the fact that the lengths of the various strings don't change (also known as "conservation of string").
It may seem, with the angst it can bring,
That an Atwood's machine's a harsh thing.
But you just need to say
That F is ma,
And use conservation of string!
Galileo's experiment worked because the air is sufficiently thin. Who knows what he would have concluded if we lived in a thicker medium...
What would you have thought, Galileo,
If instead you dropped cows and did say, "Oh!
To lessen the sound
Of the moos from the ground,
They should fall not through air but through mayo!"
The classic example of the independence of the x- and y-motions in projective motion is the "hunter and monkey" problem. In it, a hunter aims an arrow at a monkey hanging from a branch in a tree. The monkey, thinking he's being clever, tries to avoid the arrow by letting go of the branch right when he sees the arrow released. The unfortunate consequence of this action is that he will get hit, because gravity acts on both him and the arrow in the same way; they both fall the same distance relative to where they would have been if there were no gravity. And the monkey would get hit in such a case, because the arrow is initially aimed at him.
If a monkey lets go of a tree,
The arrow will hit him, you see,
Because both heights are pared
By a half gt2
From what they would be with no g.
Induction is a wonderful tool, but it shouldn't be abused...
"To three, five, and seven, assign
A name," the prof said, "we'll define."
But he botched the instruction
With woeful induction,
And told us the next prime was nine.
What really happened on that hill...
In Boston, lived Jack, as did Jill,
Who gained mgh on a hill.
In their liquid pursuit,
Jill exclaimed with a hoot,
" I think we've just climbed a landfill!"
While noting, "Oh, this is just grand,"
Jack tripped on some trash in the sand.
He changed his potential
To kinetic, torrential,
But not before grabbing Jill's hand.
Newton's law of gravitation holds over a vast range of distance scales. Apples, moons, and stars all behave in the same manner, dependent on the 1/r form of the gravitational potential.
Newton said as he gazed off afar,
"From here to the most distant star,
These wond'rous ellipses
And solar eclipses
All come from a 1 over r."
The realization was nice, but then it took him another 20 years to develop calculus so that he could prove it mathematically.
Newton looked at the data, numerical,
And then observations, empirical.
He said, "But, of course,
We get the same force
From a point mass and something that's spherical!"
Rockets make use of conservation of momentum, by ejecting small particles with very large speed. You never know when this technique might come in handy.
Roger Clemens was stuck on a lake,
With no wind, oars, or fuel — big mistake!
So he thought like a rocket,
And emptied his pocket,
And left all his change in his wake.
It's no coincidence that physicists study the harmonic-oscillator potential, kx2/2, so much. A quick application of the Taylor-series expansion shows that any potential looks basically like a quadratic, if you look at a small enough region around an equilibrium point.
A potential may look quite erratic,
And its study may seem problematic.
But down near a min,
You can say with a grin,
"It behaves like a simple quadratic!"
Given a sufficiently long lever-arm, you can produce an arbitrarily large torque. This fact led a well-known mathematician from long ago to claim that he could move the earth if given a long enough lever-arm.
One morning while eating my Wheaties,
I felt the earth move 'neath my feeties.
The cause for alarm
Was a long lever-arm,
At the end of which grinned Archimedes.
On a rotating platform, the Coriolis force always points in the same direction relative to the direction of motion. Whether it's to the left or to the right depends on the direction of rotation. But given w , you're stuck with one or the other.
On a merry-go-round in the night,
Coriolis was shaken with fright.
Despite how he walked,
'Twas like he was stalked,
By some fiend always pushing him right.
The kinetic energy of a body can be found by treating the body as a point mass located at the CM, and then adding on the kinetic energy of the body due to the motion relative to the CM.
To calculate E, my dear class,
Just add up two things, and you'll pass.
Take the CM point's E,
And then add on with glee,
The E 'round the center of mass.
t =dL/dt is valid only if the origin (the point around which t and L are calculated) satisfies one of the following conditions: (1) The origin is the center of mass, (2) The origin is not accelerating, or (3) The acceleration the origin is parallel to the vector from the origin to the center of mass. This third condition is rarely invoked. So, when choosing an origin, just play it safe and heed the following...
For conditions that number but three,
We say, "Torque is dL by dt."
But though they're all true,
I'll stick to just two;
It's CM's and fixed points for me.
If a force is always applied at the same position relative to the origin around which the angular momentum is calculated (as in the case for a quick blow to an object), then D L is proportional to D p, with the constant of proportionality being the lever-arm of the force. Even if we don't know what D L and D p are, we know what their ratio is.
What L was, he just couldn't tell.
And p? He was clueless as well.
But despite his distress,
He wrote down the right guess
For their quotient: the lever-arm's l.
When studying central forces, conservation of angular momentum is utilized to write the angular dependence in terms of the radius, thereby reducing the problem to an ordinary 1-dimensional one, with a modified "effective" potential.
When using potentials, effective,
Remember the one main objective:
The goal is to shun
All dimensions but one,
And then view things with 1-D perspective.
The L2/2mr2 term in the effective potential is sometimes called the angular momentum barrier. It has the effect of keeping the particle from getting too close to the origin.
As he walked past the beautiful belle,
The attraction was easy to tell.
But despite his persistence,
He was kept at a distance
By that darn conservation of L.
When solving an F=ma differential equation, sometimes you want to write a as dv/dt, and sometimes you want to write it as v dv/dx. One of these generally works much better than the other.
a is dv by dt.
Is this useful? There's no guarantee.
If it leads to, "Oh, heck!"'s,
Take dv by dx,
And then write down its product with v.
Linear differential equations have the extremely important property that the sum of two solutions is also a solution.
For equations with one main condition
(Those linear), we give you permission
To take you solutions,
With firm resolutions,
And add them in superposition.
The standard method for solving differential equations is to simply guess an exponential solution. It may seem cheap, but it works.
This is our method, essential,
For equations we solve, differential.
It gets the job done,
And it's even quite fun.
We just try a routine exponential.
P= r gh
Larry Lobster crawls deep in the sea,
Where the pressure and depth guarantee
That all the frustrations
Of mighty crustaceans
Won't help when they have to go pee.
The minimal-action (more precisely, the stationary-action) method for solving problems yields all the same results as the standard F=ma method, but it avoids the use of force. You simply have to write down the difference between the kinetic and potential energies, and then take some derivatives. In many cases, writing down these energies (which are scalars) proves to be much simpler than trying to write down all the forces (which are vectors), which may be pointing in all sorts of complicated directions.
It just stood there and did nothing, of course,
A harmless and still wooden horse.
But the minimal action
Was just a distraction.
The plan involved no use of force.
The Euler-Lagrange equations in verse:
When jumping from high in a tree,
Just write down del L by del z.
Take del L by z dot,
Then t-dot what you've got,
And equate the results (but quickly!)
It is sometimes said that nature has a "purpose," in that it seeks to take the path that produces the minimum action. In fact, nature does exactly the opposite. It takes every path, treating them all on equal footing. We simply end up seeing the path with a stationary action, due to the way the quantum mechanical phases add.
When an archer shoots an arrow through the air, the aim is made possible by all the other arrows taking all the other nearby paths, each with essentially the same action. Likewise, when you walk down the street with a certain destination in mind, you're not alone.
When walking, I know that my aim
Is caused by the ghosts with my name.
And although I don't see
Where they walk next to me,
I know they're all there, just the same.
Symmetries and conservation laws go hand in hand...
As Noether most keenly observed
(And for which much acclaim is deserved),
We can easily see
That for each symmetry,
A quantity must be conserved.
The "right-hand" rule for calculating vector cross-products and such things is just a convention. You would get all the same answers to any physical questions if you (consistently) used the "left-hand" rule.
When crossing your vectors at school,
You'll use your right hand as a tool.
But look in a mirror,
And then you'll see clearer,
You could just use the "left-handed" rule.
They searched and searched, but they just couldn't find it...
The findings of Michelson-Morley
Allow us to say very surely,
" If this ether is real,
Then it has no appeal,
And shows itself off rather poorly."
Everything is relative - there are no special places in the universe. We gave up having the earth as the center, so let's not give any other point a chance, either.
Copernicus gave his reply
To those who had pledged to deny.
"All your addictions
To ancient convictions
Won't bring back your place in the sky."
Perhaps the most fundamental effect of relativity is the loss of simultaneity. Two events that are simultaneous in one frame are not necessarily simultaneous in another.
Of the many effects, miscellaneous,
The loss of events, simultaneous,
Allows A to claim
There's no pause in B's frame,
Where this last line is not so extraneous.
Relativistic length contraction is a strange thing, indeed...
Relativistic limericks have the attraction
Of being shrunk by a Lorentz contraction.
But for readers, unwary,
The results may be scary,
When a fraction . . .
As is time dilation...
The effects of dilation of time
Are magical, strange, and sublime.
In your frame, this verse,
Which you'll see is not terse,
Can be read in the same amount of time it takes someone else in another frame to read a similar sort of rhyme.
The reason why those pesky muons reach the earth depends on your point of view. What time dilation explains in the earth frame, length contraction explains in the muon frame.
Observe that for muons, created,
The dilation of time is related
To Einstein's insistence
Of shrunken-down distance
In the frame where decays aren't belated.
On relativistic velocity addition:
For a bullet, a train, and a gun,
Adding their speeds can be fun --
Take a trip down the path
Paved with Einstein's new math,
Where a half plus a half isn't one.
The fact that s2=c2t2-x2 is invariant under Lorentz transformations of x and t is exactly analogous to the fact that r2 is invariant under rotations in the x-y plane. The coordinates themselves change under the transformation, but the special combination of c2t2-x2 for Lorentz transformations, or x2+y2 for rotations, remains the same. All inertial observers agree on the value of s2, independent of what they measure for the actual coordinates.
"Potato?! Potahto!" said she,
"And of course it's tomahto, you see.
But the square of ct
Minus x2 will be
Always something on which we agree."
The concept of the "relativistic mass", g m, is an unfortunate one. The goal, apparently, of such a definition is to have a relativistic particle of mass m behave exactly like a Newtonian particle of mass g m. This goal, however, is futile. A quick look at F=dp/dt shows why. For a transverse force, you can show that F=(g m)a, which does take the desired form. But for a longitudinal force, it turns out that F=(g3m)a, which does not take the desired form.
"Force is my a times my 'mass',"
Said the driver when starting to pass.
But from what we've just learned,
He was right when he turned,
But wrong when he stepped on the gas.
Results from relativistic physics must of course reduce, in the nonrelativistic limit, to the good old results from Newtonian physics.
Whether abstract, profound, or just mystic,
Or boring, or somewhat simplistic,
A theory must lead
To results that we need
In limits, nonrelativistic.
...And here's how one particular law reduces:
They said, "F is ma, bar none."
What they meant wasn't quite as much fun.
It's dp by dt,
Which just happens to be
Good ol' ma when g is 1.
Any quantity with a few factors of c is bound to change the face of the world.
The power of m's and c-squares
Provides us with just cause for scares.
Our childhood fright
Of a bump in the night
Is now mushrooms bequeathed to our heirs.
Just as we must use only vectors to describe a theory that is invariant under rotations, so must we use only 4-vectors to describe a theory that is invariant under Lorentz transformations.
God said to his cosmos directors,
" I've added some stringent selectors.
One is the clause
That your physical laws
Shall be written in terms of 4-vectors."
In an accelerated reference frame, it is possible to hang on to Newton's F=ma law, but only if you introduce new "fictitious" forces. Although these are not true forces, it impossible to perform a local experiment that differentiates these from the actual forces. This fact led Einstein to his General Theory of Relativity.
As Einstein explored elevators,
And studied the spinning ice-skaters,
He eyed as suspicious,
The forces, fictitious,
Of gravity's great imitators.
As can be shown using the equivalence principle that high clocks in a gravitational field run fast. A twin from Denver will have aged more than a twin from Boston. True, only by a fraction gh/c2 , but you never know who might notice...
Greetings! Dear brother from Boulder,
I hear that you've gotten much older.
And please tell me why
My lower left thigh
Hasn't aged quite as much as my shoulder.
The cosmological constant. "The biggest blunder of my life," he said...
Though Einstein's equations were firm,
There was one thing that did make him squirm.
A cosmos expandable
Was not understandable,
So he tacked on that ill-fated term.
The universe is expanding, but will it do so forever? The question is open. (But the answer may not be.)
The cosmos according to Hubble
Expands like the soap of a bubble.
Let's hope it's not closed,
It would then be disposed
To shrink to a point, and that's trouble.
With our new-found abilities of the late 20th century, we've started searching for friends out there. Why? Because we can. It never hurts to look under the lamppost. It's just a very big one in this case.
As we grow up, we open an ear,
Exploring the cosmic frontier.
In this coming of age,
We turn in our cage,
All alone on a tiny blue sphere.
There's nothing like Rayleigh scattering to make the sky blue...
A child looked up in the sky,
And said, "It's so blue, Mom, but why?"
Well, blue scatters more
(There's this power of 4),
So it rarely comes straight to your eye.
The second law of thermodynamics...
Humpty, he sat on a wall,
Then Humpty, he had a great fall.
But all the kings horses
And men with their forces
Couldn't render his entropy small.
On the uncertainty principle:
An electron is sure hard to please.
When spread out, it sometimes will freeze.
It's still claustrophobic,
And runs off when put in a squeeze.
In quantum mechanics, observables are represented by Hermitian matrices, with the observed value being an eigenvalue of the matrix. Hermitian matrices have nice property that their eigenvalues are real. This is very fortunate, of course, since no one is going to go out for a jog of 6+9i miles, or pay an electric bill for 17-42i kilowatt-hours.
God's first tries were hardly ideal,
You see, complex worlds have no appeal.
So in the present edition,
He made things Hermitian,
And this world, it seems, is quite real.
Progress without prudence...
As acid rain falls on the leaves,
And Mother Earth quietly grieves,
These blasted pollutants
Will make us all mutants,
And our kids will wear coats with three sleeves.
In science, we test a theory by performing experiments. However, all an experiment can show is that the theory is either consistent with experiment, or wrong. If the theory is consistent, then odds are that it's actually not true, but simply the limiting case of a more correct theory. That's how science works. You can't prove anything, so you learn to settle for the things you can't disprove.
Consider, when seeking gestalts,
The theories that science exalts.
It's not that they're known
To be written in stone.
It's just that we can't say they're false.
And speaking of limiting cases of more correct theories...
There once was a classical theory
Of which quantum disciples were leery.
They said, “Why spend so long
On a theory that’s wrong?”
Well, it works for your everyday query!
Brains are great for doing physics, but a little luck now and then certainly doesn't hurt. Besides, who will ever know the difference...
Professor, you should be commended
On your theory so geniusly splendid.
But some say it's luck,
And you really just suck,
'Cause your theory's not what you intended!
© 2004 by David Morin