# What is the physics problem whose answer is $e$ in the power of $\pi$ despite all given numbers being unity?

A few years ago my school physics teacher told us, his students, that there is an elegant physics problem in which all given numbers are unity, but the answer is $$e$$ in the power of $$π$$, that is,

$$e^\pi$$,

where e = 2.718... is the base of the natural logarithm and π = 3.14... is pi, the constant commonly defined as the ratio of a circle's circumference to its diameter.

Fast forward to now, I am a university student studying something unrelated to physics, and I recently mentioned the above recollection of mine in a conversation with a physicist whom I had helped improve his English in his physics articles. Long story short, he highly doubts that such a physics problem exists. He believes that either I misunderstood my school teacher or the formulation of the problem is too lengthy or unnatural. But I clearly remember my teacher's words that the formulation of the problem is simple and that the solution is elegant. Unfortunately, I do not know what that problem is about. I tried googling, but did not get any lead.

What is that mysterious problem, or is there any physics problem matching the description above?

• Did you encounter complex numbers when your school physics teacher mentioned this number? – R. Romero Feb 26 at 19:04
• @R.Romero : He mentioned the existence of such a problem very briefly and as a side remark unrelated to the subject of that particular lesson, not even telling us what the problem is about. As far as I remember, no complex numbers were discussed or mentioned during that lesson or in relation to that remark. I am sure that I recall the remark right, i.e., the answer is e in the power of π and does not contain i. – Mitsuko Feb 26 at 19:48

$$e^\pi$$ is the square root of the ratio of the load tension to the hold tension for a rope wrapped $$1$$ full turn around a capstan when the coefficient of friction is $$1$$.

If I am allowed to say “$$1$$ half-turn” without violating the “unity” requirement then taking the square root is unnecessary.

The relationship is

$$T_\text{load}=T_\text{hold}\,e^{\mu\phi}$$

where $$\mu$$ is the coefficient of friction and $$\phi$$ is the wrapping angle, as explained in the Wikipedia article “Capstan equation”.

• Doesn't saying "half a turn" instead of "full turn, divided by 2" make the formulation more elegant—and more natural? After all, when both ends of a rope hang from the sides, it's quite a natural position. – Ruslan Feb 26 at 18:04
• @Ruslan I have revised the answer to meet the requested criterion of “all given numbers being unity”. – G. Smith Feb 26 at 18:09
• @Ruslan we want to maximise the number of unities in the question. :P – Superfast Jellyfish Feb 26 at 18:09

The question posted in the existing answer is almost certainly the one your teacher was referring to. For the sake of illustration I'll give another question that works.

There are $$N$$ identical tiny discs lying on a table with total mass $$M$$. Another disc of mass $$m$$ is very precisely aimed to bounce off each of the discs exactly once, then exit opposite the direction it came. Neglect any collisions between the discs themselves. In the limit $$N \to \infty$$, what is the minimal value of $$M/m$$ for this to be possible? Given this value, what is the ratio of the initial and final speeds of the disc?

The answer to this question is $$e^{\pi}$$. Notably, you don't even need to set any adjustable parameters to one by hand, like you have to set $$\mu = 1$$ in the capstan problem. In this problem, $$M$$ is arbitrary, $$N$$ is taken to infinity, and $$m$$ is fixed given $$M$$.

The link between these problems is that the $$e$$ comes from some sort of exponential decay, while the $$\pi$$ comes from the fact that there are $$\pi$$ radians in a semicircle. (The problem is solved by considering the angle through which the velocity of the mass $$m$$ must turn, which is $$\pi$$.)

• What if we do not require the $N$ identical discs to lie on a semi-circle? Namely, if we only require that the incoming disc ends up going in the opposite direction, and minimize the mass of the $N$ identical discs, is the ratio of the initial to final speed of the disc still $e^π$ as $N → ∞$? If not, how can we weaken the original condition so that the answer is $e^π$? – user21820 Mar 31 at 9:22
• @user21820 Yes, that's correct, the discs don't have to be on a semicircle at all, and the answer remains the same. The key to solving the problem is to note that each individual collision can only turn the angle of the velocity by order $m/(M/N)$. So this doesn't depend on the global arrangement of the discs in space. – knzhou Mar 31 at 19:47

Incomplete playing detective. Not a physical example, but a mathematical one from which it might be possible to generalize to several physical problems.

If you have a rate of change of some quantity proportional to that quantity, i.e. $$\frac{dx}{dt}=kx$$, then $$x=c_0e^{kt}$$, where $$c_0$$ is the initial value of x.

If $$c_0=1$$ then one second later, $$x=e^\pi$$.

So what could make $$k=\pi$$? $$k$$ can be the arc length of a semi-circle, the area of a unit circle, or the volume of a cylinder having unit radius and height.

So if a rate of change is proportional to the present value of the quantity and one of these physical characteristics we should have $$e^\pi$$.

Area is a bottle neck for several processes. Chemical reactions, bacteria growth, heat transfer.