Tether Ball

Our local swimming pool had a game called tether ball, where a ball was attached with a rope to a pole, and each of the two players attempted to wrap the rope around the pole until the ball came to a stop.  The winner got to choose the wrapping direction, an advantage given one's handedness.

The other night, a former graduate student of mine posted the problem of the dynamics of motion of a tethered ball, undoubtedly leading many readers to a sleepless night.  It's called nerd sniping; us nerds have to drop everything in a compulsive search for a solution.

The figure shows the problem, where we ignore gravity and search for a solution for the motion of the ball as it wraps around the pole of radius $R$, whose size I have exaggerated for clarity.  The ball of mass $m$ is assumed to be a point and is attached to a massless rope of length $L$.  The rope is attached at point $A$ and the angle $\theta$ to point $B$ determines the part of the rope in contact with the pole.  We chose the center of the pole as the origin of the inertial frame from which we will calculate the equations of motion. 

Physical Reasoning

A physicist can often solve a problem quickly using intuition based on experience.  For example, in this case, we can imagine that for an infinitesimal time, the rope can be viewed as pivoting about point $B$, so the tension in the rope $T$, which pulls the mass toward point $B$, is always perpendicular to the velocity $v$.  As such, the work done on the mass must vanish so the kinetic energy is constant.  Then, the velocity remains constant.  On the other hand, the angular momentum decreases.  If the initial velocity of the mass is $v_0$ at point $A$, the angular momentum is $m v_0 L$.  When the rope has gone around once so that its length is now $L - 2 \pi R$, if the velocity is the same, the angular momentum is $m v_0 (L - 2 \pi R)$.  Upon each wind, the mass loses angular momentum $2 \pi m v_0 R$.

Where does this angular momentum go?  If the pole were were free to rotate upon it's axis, the pull of the rope at contact point $B$ would apply a torque, so the angular momentum lost by the mass would be transferred to the pole.  Since the pole is attached to a massive earth, it is the earth that absorbs the angular momentum.

There are many things about this problem that make me uneasy about such reasoning, and some of it stems from those times that I have been fooled by a subtlety.  For example, you might have falsely argued that angular momentum must be conserved since the pole is obviously fixed and only action and reaction forces are acting.  Alternatively, you might be concerned with the energy conservation argument because the contact point is moving around the surface of the pole, and is therefore not an inertial frame.  The key to why no work is done by the rope is the that the velocity and tension are always perpendicular in any frame.

If you are uncomfortable with using intuition, the Lagrangian method is for you.  Not only is it fool proof, but this problem is easy to solve and has some neat properties.

Lagrangian Method

From the figure, the coordinates of the mass are $x = R \cos \theta  - (L-R \theta) \sin \theta$ and $y = R \sin \theta  + (L-R \theta) \cos \theta$ which gives $r^2 = R^2 + (L - R \theta)^2$, as expected.

The Lagrangian contains only the kinetic energy terms and after some simple trigonometric identities leads to the beautiful result $${\cal L} = \frac {1} {2}  m (\dot{x}^2 + \dot{y}^2  )= \frac {1} {2} m (L-R \theta)^2 \dot{\theta}^2.$$ The Euler Lagrange equation yields $$(L - R \theta)^2 \ddot{\theta} = R(L-R \theta) \dot{\theta}^2,$$ and with the simple substitution $\ell = L - R \theta$, with $\ell$ being the unwound part of the rope, gives $$\ell \ddot{\ell} + \dot{\ell}^2 =0,$$ which gives the exact differential $$\frac {d} {dt} (\ell \dot{\ell}) = 0.$$   Integration gives $\ell \dot{\ell} = c$, yielding the general solution $$\ell^2 = 2 c t + b,$$ where $b$ and $c$ are integration constants that we fix by setting $\ell = L$ and $\dot{\ell} = \dot{\ell}_0$ at $t=0$ to give $$\ell =  \sqrt{L^2 + 2L \dot{\ell}_0 t}.\hspace{3em} (1)$$

At this point, we are done but it is fun to recast the results in other ways.  We can re-express the Lagrangian in terms of the length of unwound string, which gives $${\cal L} = \frac {1} {2} m \frac {\ell^2} {R^2} \dot{\ell}^2, \hspace{3em} (2)$$ so the velocity is given by $$v = \frac {\ell} {R}  \dot{\ell} \hspace{3em} (3)$$ yielding an initial velocity of $$v_0 = \frac {L} {R}  \dot{\ell}_0. \hspace{3em} (4).$$   Combining Equations 1 and 4 we get $$\ell = \sqrt{L^2 + 2 R v_0 t},$$ and Equation (3) gives $v = v_0$, confirming our intuition that energy is conserved. 

Conclusion

This result allows us to calculate many other things.  For example, what is the period?  How much time does it take for the rope to coil?  What is the total distance traveled by the mass?  What is the equation of the orbit?  I'll let you lose some sleep over this.  I'm not sure why I spent so much time playing with this problem given all the other stuff that is on my plate, but I got sniped and wanted to check my reasoning with the Lagrangian.  Now I'm moving on to my real work.  Feel free to point out errors in my logic.  

Meeting the Snob Factor

The best journals employ a snob factor as a first cut to limit the deluge of submitted manuscripts that go out for peer review.  The editor uses the "desk reject" for potential papers that don't look interesting.  Then, the reviewers are asked to evaluate a manuscript's significance to the field prior commenting on the technical details.  These two layers of subjective assessment can doom a manuscript, relegating it to a polite rejection: the work might be technically correct, but it is not of broad enough interest.

One such journal is Optics Letters, published by the Optical Society of America.  Though it is eclipsed by the new OSA journal Optica in its impact, it is still a highly selective and respectable publication.  Recently, we beat the odds by receiving an acceptance letter (subject to minor revision) along with the initial reviews.  The preprint of the paper can be viewed at https://arxiv.org/pdf/1809.01216.pdf

While the paper is based on some esoteric principles, it provides the experimentalist with a recipe for adding one state to the simple model commonly used in the field to correct for the infinite number of states that are omitted for bovious practical reasons.  This magical state is a proxy for those infinite numbers of states that are ignored.  The figure shows a plot corresponding to the uncorrected model (left) and the corrected one (right).  The nice smooth green background and the sharp red along the diagonal is the signature of success.  We thought it cool and useful that such a proxy state could fix a problem that has been plaguing nonlinear-optical measurements for decades.  For once, the editor and reviewers agree.

Here is a summary of the reviews:

Reviewer 1:

The manuscript represents an important advance in the calculation of nonlinear susceptibilities because it presents for the first time a method for dealing with the difficult continuum states present in realistic models of molecules. Ignoring these states leads, as the authors identify, to large errors in the calculations while, perhaps surprisingly, a single proxy state allows one to eliminate these errors to a large degree. This proxy state is not just a mathematical fudge, it is defined through physically measurable quantities. I therefore strongly recommend publication. 

Reviewer 2:

This is an interesting work discussing corrections to polarizability and hyperpolarizability calculations for limited state models that can be made using a single proxy state.  The conclusions are well supported by the calculations and this will find significant interest in its community.  

Perseverance trumps intelligence

I have not posted here for almost a year.  Perhaps it's because I have fallen into a bit of depression.  On the bright side, I have been thinking about neat things and trying to build a deeper understanding on topics where everyone else has failed.  So most likely, I will fail too, which is not of great concern to me; rather, I feel that I have not been directing my energies into areas that will most certainly bear fruit, so I might be left in the dust as a result.  I can already see evidence that I am losing connections to my colleagues.

I am concerned that this depression is affecting many aspects of my job.  Reading an article earlier today that was published prior to me presenting the Distinguished Faculty Address in 2005 (click here for the link), I felt that the same narrative would not apply to me now because of this dark undercurrent that draws away energy.

This morning I got a glimmer of hope when I asked my 20-student mechanics class why they chose to major in Physics.  To fill the elongated pause, I offered, "To gain an understanding of how things work?  To use your knowledge to invent things?  To get rich?  To be poor?  You can do all these things with a degree in physics."  This generated a bit of laughter.  I then continued, "for me, I pictured myself being dirt poor, and living in a cave with a blackboard and some physics books.  I sold out and took a lucrative industry job at Bell Labs, which I disliked.  It wasn't that the work was uninteresting --  I ended up doing lots of fun physics, but I disliked the focus on a product."

I then went on, "However, I find it fun to make things and it was very enjoyable being part of a startup that I founded two decades ago.  We did everything from basic research, to making products, and it felt a lot like academics.  There was no boss telling us what to do.  We all did what needed to be done.  That company is now highly successful, but unfortunately, I got out of it years back before it took off."

Though the story brought back fond memories, I quickly got back into the subject matter.

Later in the day, I got an email from a student that said my question resonated with him and he described his passions and how a degree in physics would get him to his ideal career.  However, he admitted that the subject was difficult to him and that he was struggling.

My reply to him (edited to keep anonymity) :


Dear so and so,

A very famous and highly respected colleague of mine (and more senior than I am) once confided in me that he feels like a fraud.  Everything he does seems to take much more effort than it does to others and he feels barely afloat.  The well kept secret is that many people feel this way, especially the successful ones because it drives them to work to exhaustion.  Your approach is spot on; work hard to realize your dream.  And don't be concerned about exam scores; I  consider exams to be a learning experience, so try to redo the problems and learn from your mistakes.  I will include a part of a problem from this exam on the next one to make sure you have learned form the process.

I always get myself into positions where I am struggling, because being challenged is a sign that you are learning and growing.  So relish the feeling that you are struggling and that you might at times feel stupid;  that's a good thing.  Check out https://www.improbable.com/2011/04/29/the-importance-of-stupidity-2/.  I also wrote on the topic a couple years back.  Take a look at https://unknownphysicist.blogspot.com/2016/11/i-always-feel-stupid-and-confused.html.
Bottom line is that I never worry about students like you.  I can see that you are attentive in class, that you are following the material, and that you participate when I ask questions.  Keep the passion!

Best,

I hope that this letter has sparked the beginning of a return to my previous self.  As I have found, writing this blog has therapeutic value even if it is never read.  So more posts in the future will be a metric of my success in overcoming my malaise.

The next topic for discussion may be my feeling of mental deterioration.  A faculty job requires multitasking at many levels, and I feel that in my older more feeble age, being diverted from actively practicing physics drains creative energy and dulls my senses.  That's something that needs to be reveresed.

I still get excited when I learn something new, and research continues to be stimulating.  The power of physics to explain phenomena, which can be harnessed to the betterment of humanity, is what attracted me to physics, and living with it throughout my lifetime has deepened my love for it.

That's a good starting point to build on...