Math Grenade

I just bought a new calculator. New to me, that is, but older than me.

Inspired by the 1600s Gottfried Leibniz stepped cylinder and the 1800s Thomas de Colmar arithmometer, the Curta mechanical calculator design was developed by Curt Herzstark while imprisoned in the Buchenwald concentration camp during World War II. Curta calculators were manufactured between 1947 to 1972 in Liechtenstein until electronic calculators became widely available.

A true number cruncher, the Curta has been nicknamed the pepper grinder because of its crank — or the math grenade due to its resemblance to some hand grenades. My Curta II has about 719 metal parts yet fits comfortably in my hand. I remember seeing advertisements for Curta calculators in Scientific American magazine. Now collector’s items, the one I just bought costs an order of magnitude more.

“The world’s first, last, and only hand-held mechanical calculator”, the Curta is an elegant marvel of mechanical engineering, the culmination of generations of work to mechanize arithmetic. And as I can now attest, it is a joy to hold and use.

After 2 slides and 2 cranks, a Liebniz wheel computes 5 + 3 = 8

A Liebniz wheel at the heart of the Curta computes 5 + 3 = 8 with 2 slides and 2 cranks

Curta ad from my childhood

Curta ad from my childhood

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Optical Tweezers

A focused light beam can trap a small particle, such as a micron-sized latex sphere (or biological cell). If the sphere is much larger than the light’s wavelength, ray optics suffices to explain the trapping.

Light bends as it passes through the sphere, as in Fig. A. The piconewton forces (red and blue arrows) on the sphere are equal and opposite to the rate of momenta transferred to the light rays: the rays deflect one way, and the sphere deflects the opposite way. The net force (black arrow) is longitudinal and pushes the bead along the beam. A beam that is less intense at its edge nudges a laterally offset sphere back towards its center, as in Fig. B. A focused beam pulls a sphere behind the focus toward the focus, as in Fig. C. The push in Fig. A stabilized by the nudge in Fig. B balances the pull in Fig. C to hold the sphere just behind the focus, as in Fig. D: a Nobel Prize hiding in plain sight.

Microsphere in a light beam: Fig. A push stabilized by Fig B nudge balances Fig. C pull to trap sphere

Microsphere in a light beam: Fig. A push stabilized by Fig B nudge balances Fig. C pull to trap sphere

A single Nobel Prize cannot be awarded to more than three people and cannot be awarded posthumously. Arthur Ashkin developed optical tweezers in the 1970s and 1980s. Over 20 years ago his collaborator Steven Chu shared the Nobel Prize with Claude Cohen-Tannoudji and William Phillips for trapping atoms. This year at age 96 Ashkin receives the Nobel Prize for his optical tweezing pioneering.

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Dr. Rendezvous

Edwin Aldrin earned his PhD from MIT in 1963 with a thesis titled, “Line-of-sight guidance techniques for manned orbital rendezvous”. Just three years later in 1966, Aldrin was the pilot of Gemini XII, the last flight of the Gemini program, a critical precursor to the Apollo moon program. Aldrin and his commander James Lovell were attempting to rendezvous and dock with an Agena target vehicle when their onboard radar failed. Aldrin used a handheld sextant to repeatedly measure the angles between Gemini XII and the Agena ahead and above them to help Lovell successfully complete the rendezvous and docking.

Aldrin would join Neil Armstrong and Michael Collins on the historic flight of Apollo 11 in 1969 — and later formally change his first name from “Edwin” to his other nickname “Buzz”.

When Gemini XII's radar failed, the man who wrote the book on rendezvous was onboard to help

When Gemini XII’s radar failed, the man who wrote the book on spacecraft rendezvous was onboard to help

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Ancient cultures everywhere observed seven “wanderers” move against the apparently fixed stars of the night sky: our star the sun, our natural satellite the moon, and the brightest planets Mars, Mercury, Jupiter, Venus, and Saturn. In many languages, these wanderers became the basis for the names of the seven days of the week; for example, in English & French the days are:

Sunday & dimanche
Monday & lundi
Tuesday & mardi
Wednesday & mercredi
Thursday & jeudi
Friday & vendredi
Saturday & samedi

In many languages, the days of the week are named after the classic "planets"

In many languages, the days of the week are named after the classic “planets”

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A falling cat’s twisting returns its shape to normal but rotates its body to land feet down. Earth’s spin returns a Foucault pendulum to its initial position in one day but rotates its oscillation plane. Parallel parking cyclically rotates a car’s front wheels but shifts the car sideways. These are examples of nonholonomic motions or mechanical anholonomies.

Hwan Bae, Norah Ali, and I just published a featured article in the journal Chaos on another famous anholonomy, Hannay’s hoop, which involves a bead sliding frictionlessly on a horizontal noncircular hoop: A slow cyclic rotation restores the hoop to its original state but unavoidably shifts the moving bead by an angle that depends on the hoop’s geometry. Rotating a noncircular hoop indelibly imprints its geometry on the bead’s motion.

In the limit of slow rotation and fast beads, the shift is called Hannay’s angle (and is analogous to Berry’s phase in quantum mechanics). We mathematically generalized the shift to any speed, fast or slow, and were able to observe it in a simple experiment involving wet ice cylinders sliding in 3D printed channels.

Spacetime diagrams of a bead sliding on spinning hoop in the hoop (left) and lab (right) frames. Red spheres are experiment, cyan tubes are simulation, yellow tubes numerically extrapolates bead sliding in the absence of hoop spinning, and black arcs are the generalized Hannay’s shift.

Spacetime diagrams of a bead sliding on spinning hoop in the hoop (left) and lab (right) frames. Red spheres are experiment, cyan tubes are simulation, yellow tubes numerically extrapolates bead sliding in the absence of hoop spinning, and black arcs are the generalized Hannay’s shift.

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Norton’s Dome

Norton’s Dome is a fascinating counterexample in classical mechanics: A frictionless mass balanced at the dome’s top can remain there forever — but can also spontaneously slide down!

Mass at peak of Norton’s Dome waits infinitely long and then spontaneously slides to the bottom. Carefully kicking the mass back up causes it to stop at rest at the peak after a finite time. Since rest at top is a solution, why does it ever slide down?

A mass slides down Norton’s dome from rest and then is kicked back up to rest.

The Shape

Norton’s dome has a cubed square root profile: if the downward height is z at dome arc length s, then

\frac{z}{z_m} = \left(\frac{s}{s_m}\right)^{3/2}.

If max height z_m = 2/3 when max arc length s_m = 1, then

z = \frac{2}{3}s^{3/2}.

By the Pythagorean theoremds^2 = dx^2 + dz^2, and so

x = \int dx = \int \sqrt{ds^2 - dz^2} =\int ds \sqrt{1 - \left(dz/ds\right)^2} = \int_0^s ds \sqrt{1 - s}.

Integrate to find the horizontal coordinate in terms of the arc length

x = \frac{2}{3}\left(1 - \left(1-s\right)^{3/2}\right),

and inversely

s = 1 - \left(1 -\frac{3}{2}x\right)^{2/3}.

Eliminate the arc length to find the profile

x =\frac{2}{3}\left(1 - \left(1-\left(\frac{3}{2}z\right)^{2/3}\right)^{3/2}\right),

and inversely

z =\frac{2}{3}\left(1 - \left(1-\frac{3}{2}x\right)^{2/3}\right)^{3/2}.

If the arc length 0\le s \le 1, then the rectangular coordinates -2/3 \le x,y \le 2/3 and 0\le z \le 2/3.


For sliding frictionlessly on the dome, form the total energy as the sum of the kinetic and potential energies

E = T + V = \frac{1}{2} m v_s^2 - m g z = \frac{1}{2}\dot{s}^2 - \frac{2}{3} s^{3/2},

for unit mass and gravity. Differentiate with respect to time to get

0 = \dot{s} \ddot{s} - s^{1/2}\dot{s}

and divide by \dot s to get the equation of motion

\ddot{s} = \sqrt{s}.

Alternately, form the Lagrangian as the difference in the kinetic and potential energies

L = T - V = \frac{1}{2} \dot{s}^2 + \frac{2}{3} s^{3/2}

and insert into the Euler-Lagrange equation

\frac{d}{dt} \frac{\partial L}{\partial \dot{s}} = \frac{\partial L}{\partial s}

to again get

\ddot{s} = \sqrt{s}.

The acceleration along the dome increases as the square root of the arc length. Combine the equation of motion with the initial conditions s[0]=0 and \dot{s}[0]=0, corresponding to a mass at rest on top of the dome, to form an initial value problem (IVP).

The Solutions

Surprisingly, two solutions exist: infinite rest

s[t] = 0

and spontaneous sliding

s[t] = \frac{1}{144} t^4.

To check the latter, differentiate to get

\dot s = \frac{1}{36} t^3, \ddot s = \frac{1}{12} t^2 = \sqrt{s}.

How can a physical system have two evolutions?

The Resolution

If an IVP’s equations are continuous, solutions exist; if the first derivatives are also continuous, the solution is unique. In this case, the square root \sqrt{s} in the equation of motion means that the first derivative is not continuous at the origin, so solutions exist but are not unique.

The peak of Norton’s Dome is continuous, and its slope is continuous, but the slope of its slope is discontinuous. The apex of a cone is more extreme, as its slope is discontinuous.



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Standard Model at 50

From Haidar Essili:

All I can think of to describe my experience in the Standard model’s 50th anniversary conference is to repeatedly yell the word wow, until I have lost the will to do so. I am at a loss of words, but I will attempt to put my flustered speech in perspective.
Imagine Albert Einstein dedicating some of his time to share his findings with you. One of the greatest minds explaining the greatest discoveries to you. Not to get greedy but imagine all the great minds of a generation dedicating a couple of days to do just that. I am of course referring to 1927 fifth Solvay International Conference featuring the most prominent names in physics; names including Albert Einstein, Niels Bohr, Paul Dirac, Erwin Schrodinger, and Wooster’s very own Arthur Compton. Attending a conference of such magnitude is a dream for any aspiring physicist, and this is how it felt to be a part of the Standard Model Conference.

Dr. David Gross and Haidar Essili during SM@50.

Dr. Gerard ‘t Hooft and Haidar Essili during SM@50.

In this conference, I listened to my idols talk about the successes of this field I aspire to be a part of. I also listened to my idols talk of the short comings of this theory. No where is the Greek quote “the more I know, the more I know I know nothing” more true than when the most knowledgeable scientists in a field tell of how little we know. In fact, the Standard Model, often dubbed the Theory of Almost Everything, accounts for a mere 4% of the universe’s constituents and even that we don’t fully understand. I sat in the back when speakers took turns calling upon the back rows of undergraduates for help in advancing this theory in the future. Imagine Einstein passing the torch to you. Imagine Einstein asserting the back rows will certainly produce the next Nobel Laureate. Wow!!”

Gerard t’ Hooft (1999) asks Steven Weinberg (1979) a question after the final talk at SM@50. Carlo Rubbia (1984) and David Gross (2004) listening (center left part of the lecture room). In Front of us were Jerome Friedman (1990) and George Smoot (2006). Samuel C.C. Ting (1976) and Takaaki Kajita (2015) left after their presentations during the weekend. In parenthesis are the years of their Nobel Prize in Physics award.

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Spring Outreach Events

Spring is a big time for outreach here at Wooster Physics. The Physics Club runs demonstrations for local elementary schools, doing often two outreach visits a week during the spring.  (In the fall, we are usually prepping for this flurry of events — sending letters to the schools and doing scheduling, and training new students on the outreach activities.)

We also have two big events on campus.  The Physics Club runs Science Day, an event for all science clubs on campus  to do demos and fun activities for the whole community.  And we participate in Expanding Your Horizons, a huge event specifically for middle school girls that incorporates not just women science students and professors from the campus but also professional women from around the community whose job includes an aspect of science.

At Science Day, it’s fun to see what the other sciences on campus are doing.  The neuroscience club gets a lot of interest with the brains that they bring.

Brains! Brain hats! Lots of brains!

Brains! Brain hats! Lots of brains!

Build your own DNA -- so popular, they had to run to the store for more gummy bears

Build your own DNA — so popular, they had to run to the store for more gummy bears

Static charge is awesome!

New this year was a giant size demo from the Astronomy Club to demonstrate how massive objects warp the spacetime around them so that other smaller objects orbit the massive one.  This was lots of fun to play with!

Spheres orbiting a massive object warping the fabric of space around it

Spheres orbiting a massive object warping the fabric of space around it

Air pressure is always a favorite, of course, with the liquid nitrogen parts. This year the demo even attracted President Bolton!  I think she had a fun morning with lots of physics — it’s probably a good change from administration.

The pink balloon slowly re-inflates as it warms back up to room temperature

The pink balloon slowly re-inflates as it warms back up to room temperature

President Bolton enjoys a little physics for her spring Saturday.

President Bolton enjoys a little physics for her spring Saturday.

Bubbling multi-colored lava lamps from chemistry.

Bubbling multi-colored lava lamps from chemistry.

Static charge is awesome!

For Expanding Your Horizons, I do the same workshop three times for different groups of girls.  We do the “Humpty Dumpty” experiment, where the girls have about 20 minutes with limited materials to create a container to try to protect an egg from breaking during a fall. We drop the eggs from the 3rd floor, so it’s pretty challenging! This year, Dr. DeGroot joined me and we had lots of fun.  I love seeing the creativity of the girls — not only in making their containers, but also in decorating and naming their eggs.

Groups of girls work on their egg containers.

Boxes? Cushioning? Parachutes? What else can we try?

Wrapping the eggs up takes lots of hands and coordination!

An anxious little egg, waiting in eggs-pectation of the fall.

An anxious little egg, waiting in eggs-pectation of the fall.

After helping with the Humpty Dumpty experiment for three year, Justine decided to try it out herself!

After helping with the Humpty Dumpty experiment for three years, Justine decided to try it out herself!

And the moment of truth — dropping the eggs from a great height! If the eggs survive, we add them to the Egg Hall of Fame!

Girls waiting 3 flights down, waiting for me to drop the eggs.

The moment of truth — will the eggs survive?

Most of the eggs this year made it! Lesson learned = parachutes really work!

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Moon Dance

I ran up the stairs to Studio Art. Justine was already rolling out the treadmill, so I climbed another flight of stairs to the old running track and let down both ends of the steel cable, one end connected to the harness and the other to the sandbag counterweight. Our reduced gravity rig was basically a giant Atwood machine leveraging technology perfected recently for flying performers in theatre and cinema. I turned on all the lights in the Crit space, and the wood floor shined.

This Saturday was our last day to work with our dancers before Justine left for LA to present her senior thesis at the March American Physical Society meeting. (From literally thousands of presentations, the APS’s PHYSICS web site would rank Justine’s Moon Dance talk as one of the meeting’s top ten highlights.)

We massed Rachel, and Justine helped her into the harness as I fine-tuned the sandbag mass to simulate lunar gravity. With Rachel sitting expectantly on the floor, Justine and I struggled to raise the sandbag and connect the steel wire to the harness with the carabiners. We moved away, and with an almost surreal lack of effort, Rachel gracefully stood, the sandbag descending as she ascended. I made a mental note to thank Mike for recommending the low-friction pulleys.

Kim and Justine had choreographed treadmill translation sequences for both Kathlyn and Rachel, but the free-dance improvisation proved most successful. Once we got the physics right, I had hoped we would produce something of artistic value, and we had. Our dancers had the grace, and we gave them the power – a superpower.

By approximating lunar and martian gravity for her senior thesis, Justine changed the physics of dance. But her central achievement was the unprecedented and dazzling reduced-gravity performances she elicited from her dancers. Later this century, dancers will dance on Luna and Mars, and Justine has glimpsed that future, and it will be spectacular.

Download a high-resolution 659 MB MP4 of our “Back to Black” video. The download may take several minutes, but it’s worth it.

Kathlyn MacDonald dancing at lunar gravity in our reduced gravity rig

A Kathlyn MacDonald floating pirouette

Rachel Lau dancing in lunar gravity in our reduced gravity rig

A Rachel Lau spinning leap

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A look back on 2017-18

Congratulations class of 2018!  I’ve attached a few pictures of some of our seniors and professors who gathered for a quick photo on graduation day (pics courtesy of Zane Thornburg).  Special congratulations are in order for Avi Vajpeyi (far left in the second photo), who was selected to be one of the College’s commencement speakers this year!

As my first year as department chair winds up, I am preparing to move to the University of Oregon in Eugene for a one-semester research leave.  I’m looking forward to reconnecting with five Wooster physics alumni who are now in the Physics Ph.D. program there!

As I look back on the year, I have a few more random highlights to post that I somehow never found the time to blog about during the busy year!

I enjoyed attending the Senior awards banquet earlier this spring, where Justine Walker, Avi Vajpeyi, and Zane Thorburg (left to right below) were honored for their academic accomplishments.  Congrats to all three of you!


The department’s new 3-D printer, housed in Dr. Manz’s Wave Lab, experienced heavy use in multiple research projects, including Collin Hendershot ’18’s project studying the aerodynamics of race car wings of various curvatures.  We also used it to 3-D print a plaque honoring Maggie Lankford 16 whose selection as an Apker Award finalist for her I.S. research  helped fund the purchase of the printer.  Thanks for Jack Mershon ’18 for his help with the plaque!

Last summer, students and professors who participated in our NSF-sponsored summer research program paused their work for an afternoon to go on a hike at Wooster Memorial park, a lovely 300 acre wooded area including many miles of trails around a large ravine. I hope this becomes an annual tradition!

The department is looking forward to another exciting summer of research as our 2018 program begins in less than one week!  I’ll be sure to blog from Oregon during my leave, so stay tuned.

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