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Recently I chewed the fat with a physics student and got intrigued by him mentioning "the Devil's problem," which he described as a simply worded mechanics problem that is extremely difficult to solve and has an answer of exactly 13 despite the formulation having no numbers and being very natural. That's kinda crazy, so I laughed and said he was kidding, but he responded that no, he wasn't kidding.

He then explained me the formulation of the problem. You've got a plastic tube, like a tube used to carry posters to conferences, but open on both ends. You firmly attach a thin but heavy rod to the inner surface of the tube, parallel to the tube's axis. The tube is then laid on a floor so that the rod is in the uppermost position, and then is released to roll away from that unstable equilibrium position. The floor isn't slippery, so there's no sliding. How many times heavier than the original tube must the rod be for the tube to jump?

I'm a student studying something unrelated to physics, and although I liked physics at school, this problem is too tough for me to crack, so I can't tell whether he was fooling me or whether what he said is true. I tried to find the problem on the Internet, but to no avail, so I'm posting it here. It's mystical and a bit scary if the Devil's dozen really pops out of nowhere in such a simply stated problem, but I guess the student was bluffing, counting on my inability to solve such problems. I don't even understand why the tube would jump.

I just made an illustration to help understand the description of the problem: enter image description here

Can the rod actually jump? If so, how can one approach this problem? Can you help me call the bluff of the student, or is 13 really the answer?

UPDATE: To clarify in response to a comment below, the rod and the wall of the tube are much thinner than the tube diameter and thus can be assumed to be infinitesimally thin. Likewise, an infinitesimal initial perturbation due to a slight asymmetry or thermal fluctuations is assumed. The problem is clearly well-posed from the mathematical standpoint, so the only question is how to solve it and what is the answer.

UPDATE 2. It seems I've figured out why the tube will jump if the mass of the rod is large enough, but I can't calculate the exact threshold. My proof of the jump is in my answer below.

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    $\begingroup$ It looks like extended discussion is getting started in the comments; I've moved it to chat. Please keep in mind that comments are meant for requesting clarification on the question or suggesting improvements to it; anything else should go to the chat room. $\endgroup$ – David Z Jun 27 at 0:20
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    $\begingroup$ Some further comments removed. The removal of "extended discussion" in comments is not an invitation to start another extended discussion about whether or not such removal was appropriate. If you have issues with moderator actions, please take it to Physics Meta. $\endgroup$ – ACuriousMind Jun 28 at 10:21
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    $\begingroup$ @ACuriousMind : >> If you have issues with moderator actions, please take it to Physics Meta << That's what I just did: physics.meta.stackexchange.com/q/12977 $\endgroup$ – Mitsuko Jun 28 at 14:21
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    $\begingroup$ Uh well can someone explain to me what exactly means jump in this case? $\endgroup$ – Richard Kiddman Jun 29 at 4:37
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I don't know if there's a beautiful solution for this. I'd love to see it, if it exists. What I can do is show you how I slogged my way through it. All praise to the mighty Mathematica.

Part I: Obtaining the Equations of Motion

First, we can dispense with the cylinder and rod and consider only a point mass $M$ on a ring of mass $m$ and radius $R$. Define $\theta$ as the angle the mass makes with the vertical, as shown:

enter image description here

All of the dynamics of this problem can be framed in terms of this angle. Assuming a no-slip condition and purely horizontal motion, the linear velocity of the center of the ring is $R\dot \theta$, where $R$ is the ring's radius. The components of the velocity of the point mass are $$v_x = R\dot \theta + \frac{d}{dt}\big(R\sin(\theta)\big) = R\dot\theta + R\cos(\theta) \dot\theta$$ $$v_y = \frac{d}{dt}R\big(1+\cos(\theta)\big) = -R\sin(\theta)\dot\theta$$

The total kinetic energy can be expressed as (i) the translational kinetic energy of the center of mass of the ring, plus (ii) the rotational kinetic energy of the ring about its center, plus (iv) the kinetic energy of the point mass. Summing all of these contributions yields

$$T = \frac{1}{2}m(R\dot \theta)^2 + \frac{1}{2}(mR^2)\dot \theta^2 + \frac{1}{2}M\left(\big(R\dot \theta + R\cos(\theta)\dot \theta\big)^2 + \big(-R\sin(\theta)\dot\theta\big)^2\right)$$ $$ = mR^2\dot\theta^2 + \frac{1}{2}MR^2\dot \theta^2\left(2+2\cos(\theta)\right)$$ $$= MR^2\dot \theta^2\left(1+\cos(\theta)+\mu\right)$$

where $\mu\equiv \frac{m}{M}$. The potential energy is simply $U=MgR(1+\cos(\theta))$, so the Lagrangian for this system is

$$L = MR^2\dot\theta^2\left(1+\cos(\theta)+\mu\right) - MgR(1+\cos(\theta))$$

and the total energy is

$$E = MR^2\dot\theta^2\left(1+\cos(\theta)+\mu\right) + MgR(1+\cos(\theta))$$

Because the kinetic part of the Lagrangian is quadratic in $\dot \theta$ and there is no explicit time dependence, $E$ is a conserved quantity. If we assume that the initial condition is an infinitesimal distance away from $\theta=0$, the total energy is equal to $2MgR$; this allows us to write

$$\dot \theta^2 = \left[\frac{1-\cos(\theta)}{1+\cos(\theta)+\mu}\right]\frac{g}{R}$$ and via differentiation, $$\ddot \theta = \left[\frac{(1+\frac{\mu}{2})\sin(\theta)}{(1+\cos(\theta)+\mu)^2}\right]\frac{g}{R}$$


Part II: The "No-Jump" Condition

The sum of the vertical components of the forces on the point mass is $$\sum F_y = F_R - Mg = M\dot v_y = -MR\big(\sin(\theta)\ddot \theta +\cos(\theta)\dot\theta^2\big)$$ where $F_R$ is the vertical component of the constraint force due to the ring. The sum of the vertical components of the forces on the ring is then $$\sum F_y = -F_R - mg + F_N = 0$$ where $F_N$ is the normal force on the ring due to the floor. The condition that the ring will never jump is that $F_N \geq 0$; this corresponds (after some algebra) to the condition $$(1+\mu)\frac{g}{R} -\sin(\theta)\ddot \theta -\cos(\theta)\dot\theta^2 \geq 0$$


Part III: Putting Things Together

We already have expressions for $\dot\theta^2$ and $\ddot \theta$; our no-jump condition becomes (dividing by $g/R$)

$$1+\mu- \left[\frac{(1+\frac{\mu}{2})\sin^2(\theta)}{(1+\cos(\theta)+\mu)^2}\right]-\left[\frac{(1-\cos(\theta))\cos(\theta)}{1+\cos(\theta)+\mu}\right] \geq 0$$

At this point, one can clearly see that there are circumstances under which the ring would jump. For $\theta=\pi+\epsilon$, the left hand side diverges to negative infinity like $-1/\mu^2$, meaning that for sufficiently small $\mu$ we can violate our no-jump condition. From here, it's a matter of rather unpleasant algebra. If you minimize the left hand side with respect to $\theta$ and grind through more algebra, the condition takes the form

$$(2+\mu)^2(13\mu-1)\geq 0$$ $$\implies \mu \geq \frac{1}{13}$$

Therefore, if $m < \frac{M}{13}$, the normal force from the floor on the ring would have to become negative; it follows that in the absence of any adhesive effects, the ring would jump up into the air.

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  • $\begingroup$ I started using Mathematica quite recently. Did you use Mathematica to do anything other than simplifying the expressions? $\endgroup$ – Archisman Panigrahi Jun 28 at 3:31
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    $\begingroup$ @ArchismanPanigrahi I used it to plot various expressions to make sure I wasn't making mistakes, and to integrate the equations of motion to verify my solutions. I made quite a few algebra errors my first time through this problem, so it was quite helpful to double-check at each stage. $\endgroup$ – J. Murray Jun 28 at 3:33
  • $\begingroup$ Your linear velocity is not that of the center of the ring, but rather the center of the rod. $\endgroup$ – Ruslan Jun 28 at 11:34
  • $\begingroup$ Thanks a lot, that's a great answer and a beautiful solution. I like it because it is easy to understand and straightforward, and your method can be used to easily solve a whole class of similar problems without much thinking. You didn't resort to tricky equations for moments of forces or angular momenta; energy conservation and the balance of vertical forces proved to be enough to obtain the answer. $\endgroup$ – Mitsuko Jun 28 at 11:46
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    $\begingroup$ @RichardKiddman The cylinder will lose contact with the ground if the normal force between the ground and the cylinder goes to zero. $\endgroup$ – J. Murray Jun 29 at 5:42
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Turns out the tube does jump if the rod is no less than 13 times the mass of the tube. My previous answer had a couple of mistakes that yielded the wrong result, here is the updated one.

Let $M$ be the mass of the tube, $m$ the mass of the rod and $R$ the radius of the tube. Let $\theta$ be the angle between the vertical and the direction of the rod from the center of the tube (i.e. $\theta=0$ initially and increases as the tube starts rolling, and the height of the rod relative to the ground is $(1 + \cos \theta)R$).

The combined moment of inertia of the tube and the rod about the instantaneous line of contact of the tube with the ground is $$I = 2R^2[M + (1+\cos \theta)m].$$

From conservation of energy, $$E = 0 = \frac{1}{2}I\omega^2 - mgR(1 - \cos \theta) $$ $$ \omega^2 = \frac{mg(1 - \cos\theta)}{R[M+(1 + \cos\theta)m]}=\frac{g\beta}{R}\frac{1 - \cos\theta}{1+(1 + \cos\theta)\beta}.$$ where $\beta=m/M$. Differentiating with respect to time, $$2\omega\alpha=\frac{d}{d\theta}\omega^2\frac{d\theta}{dt}=\omega\frac{d}{d\theta}\omega^2 $$ $$\alpha=\frac{1}{2}\frac{d}{d\theta}\omega^2=\frac{g\beta}{R}\sin\theta\frac{\frac{1}{2}+\beta}{[1+(1+\cos\theta)\beta]^2}$$

where $\alpha=d\omega/dt$ is the angular acceleration. The downward acceleration of the total system is the same as that of the rod, which is $$a_z=\omega^2R \cos\theta+\alpha R \sin\theta $$ where $\omega$ is the angular velocity. The first term is due to the centripetal acceleration and the second due to tangential acceleration. $$a_z = g\beta\frac{(1 - \cos\theta)\cos\theta + \sin^2\theta\frac{\frac{1}{2} + \beta}{1+(1+\cos\theta)\beta}}{1+(1+\cos\theta)\beta}$$ where $\beta=m/M$.

The tube remains in contact with the ground as long as the upward normal force $N$ on the tube is non-negative. By Newton's second law, $$(M+m)g-N=ma_z,$$ so the contact condition is $$N=(M+m)g - ma_z\ge0$$ $$a_z \le \left(1+\frac{1}{\beta}\right) g$$ $$ \frac{\beta^2}{1+\beta}\frac{(1 - \cos\theta)\cos\theta + \sin^2\theta\frac{\frac{1}{2} + \beta}{1+(1+\cos\theta)\beta}}{1+(1+\cos\theta)\beta} \le 1 $$

I didn't bother to do the algebra to simplify the expression on the left hand side, but I plotted the maximum of the left hand size as a function of $\beta$: enter image description here

The equality does happen at exactly $\beta=13$. Also see J. Murray's answer for the simplified condition that confirms this.

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    $\begingroup$ Puk, are you sure about your solution? If my memory serves me well, Newton's second law of rotation applies only if the rotating solid body rotates about a fixed axis so that the moment of inertia of the body about the axis of rotation always remains the same. And this condition isn't fulfilled in our case. The instantaneous axis of rotation moves, and the moment of inertia of our solid body about that axis changes with time. $\endgroup$ – Mitsuko Jun 27 at 7:20
  • $\begingroup$ @Mitsuko The instantaneous axis of rotation is actually at rest: the velocity of the bottom of the tube is always zero, since the tube is not slipping. But you are right about the moment of inertia changing, I forgot to take this into account. Newton's second law still works, there will just be another term proportional to the rate of change of the moment of inertia. I will edit my answer and correct this when I have time. $\endgroup$ – Puk Jun 27 at 8:16
  • $\begingroup$ Great, let's see whether the correction will turn the tables. Sorry for the downvote, and I'll definitely change it to an upvote after there are no obvious errors left. Thanks a lot for your effort anyway. $\endgroup$ – Mitsuko Jun 27 at 13:06
  • $\begingroup$ I've figured out why the tube will jump if the mass of the rod is large enough. I just posted my proof of the jump as a separate answer. But I still can't calculate the critical mass ratio. Perhaps you can do it if you properly take the forgotten term into account. At school, I was taught to solve rotational dynamics problems only for the case where the axis of rotation is fixed and never changes. $\endgroup$ – Mitsuko Jun 27 at 17:12
  • $\begingroup$ This is on the right track I think but there is that problem pointed out by @Mitsuko (all you need to do is use $\tau = d(I\omega)/dt $ which gives an extra term due to the changing I of the rod). There is another problem I think in that the vertical acceleration of the rod is not the same as the vertical acceleration of the center of mass. You need to use the center of mass for your argument that $a_z<g$ $\endgroup$ – octonion Jun 28 at 6:18
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It seems I've figured out why the tube will jump if the mass of the rod is large enough, but I can't calculate the exact threshold. My proof of the jump is below.

  1. Let's suppose that the mass of the original tube (i.e., without the rod) is infinitesimal, whilst the mass of the rod is finite.

  2. Let's also assume that the tube won't jump. I'm going to prove the jump by contradiction.

  3. Since we have assumed that the tube won't jump, the rod's trajectory will be a cycloid.

  4. It follows from the energy conservation law that the rod's speed at any given position will be the same as that of a small bead sliding without friction under gravity on a fixed wire of exactly the same cycloid shape, if the bead is initially in the uppermost position and has an infinitesimal initial velocity.

  5. Since the motion of the rod and the motion of the bead coincide, they are acted on by exactly the same net forces if the bead's mass is equal to the rod's mass. (If the masses are different, the difference in the net forces will be merely in the normalization coefficient, but not in the direction.)

  6. The force exerted on the bead by the wire is always directed perpendicular to the wire.

  7. Let's focus for a moment on the horizontal component of the bead's velocity. It's zero initially, in the uppermost position, then becomes finite and directed forward, but it's again zero in the lowermost position, as seen from the shape of the cycloid. Hence, the horizontal component of the acceleration of the bead changes the direction at a certain moment before the bead reaches the lowermost position. That is, starting from a certain moment before the bead reaches the lowermost position, the bead experiences horizontal deceleration.

  8. Horizontal deceleration of the bead means that the horizontal component of the force exerted on the bead by the wire is directed backwards.

  9. It follows from (6) and (8) as well as from the shape of the cycloid that during the period of horizontal deceleration of the bead, the vertical component of the force exerted on the bead by the wire is directed downwards.

  10. According to Newton's third law of motion, the bead acts on the wire with the force opposite to the force with which the wire acts on the bead. Which means that during the period of its horizontal deceleration, the bead acts on the wire with a force whose vertical component is directed upwards.

  11. In view of (5), the rod, likewise, acts on the tube with a force whose vertical component is directed upwards, during the period of horizontal deceleration of the rod.

  12. Hence, to keep the massless tube on the floor during that period, the floor must act on the tube with an attractive force. Which the floor can't do. So I have arrived at a contradiction.

Quod erat demonstrandum, as they say.

My proof also shows at which moment the tube will jump if the rod-to-tube mass ratio is infinitely large: the jump occurs exactly at the moment when the horizontal component of the rod's velocity reaches a maximum.

But calculating the critical mass ratio seems to be beyond my capabilities, so I hope someone will be able to come up with a solution.

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