Chain fountain

Fountain of the ball chain shown in the following picture. The glass is about 1.60 m from the floor. The chain rises about 10 cm above the pile at the bottom of the glass.

Ball chain on graph paper. Stretched in the straight part, the tightest possible curve in the bend. Total length about 6 m.

Play media file

Video of a chain fountain.

A chain fountain is created as a result of the mold effect when a suitable chain or rope slides down over the edge of a vessel. The falling end of the chain pulls the chain links in the vessel faster and faster. The sharp kink with which the chain initially runs over the edge of the vessel widens to form an arc that no longer touches the edge and becomes higher the faster the chain moves. The chain rises, trembling and meandering, from where it is released from the tangle. As soon as the falling end reaches the ground, the speed of the chain and the height of the fountain stop increasing, a steady state is reached. The width of the arch can be increased by tilting the vessel.

There are published video demonstrations of the phenomenon with ball chains , with a chain made of threaded macaroni (pasta a few centimeters long) and with a rope. The phenomenon first became known to the general public in 2013 through a YouTube video by Steve Mold, after which the effect was named. ^1:01 It's a discovery rarely seen in mechanics .

Physical explanation

Mold suggested an intuitive explanation about a momentum balance for the chain links currently in the air. Biggins and Warner soon showed, however, that the height of the fountain cannot be different from zero, but that in addition to the pulling force of the chain, a force pushing the starting chain piece is necessary. This force must come from the energy that the pulling force of the falling chain brings into the vessel; the development of the force in detail can depend in a complicated way on the type of chain / rope and its arrangement in the vessel.

It is easier to describe the shape of the arch from the vessel to the bottom. On average over time, it is an inverted chain line . The pulse changes direction continuously along this curve, while the amount (with the longitudinal velocity) remains constant. The vectorial impulse change is caused jointly by the gravitational acceleration on the one hand and the tensile stress in combination with the curvature on the other.

The tensile stress in the upper area of ​​the arch is approximately so great that the propagation speed of waves in the chain is equal to the longitudinal speed of the chain itself. ^{from 2:23} This explains the meanders and the tremors. These are waves that run against or with the medium, which in total results in a very low or twice the speed.

Calculation of the fountain height in the stationary state

Simplifications

• The fluctuations are not averaged, but ignored. On average, the meander would make the section of the chain in the air longer and heavier. ^{from 1:02}
• The chain line is so steep that the entire change of direction happens in a narrow apex area, the weight of which is neglected.
• When it hits the ground, the kinetic energy of the chain is dissipated without any reaction . If not, the ground level is defined by extrapolating the tensile stress to the value zero.

Tensile stress

Unconnected in free fall on a trajectory parabola , the chain links would be weightless. However, their interlinking prevents them from following the acceleration due to gravity . Starting from the value zero on the ground, the tensile stress increases linearly with height : ${\displaystyle g}$ ${\displaystyle h}$

${\displaystyle F_{\parallel }(h)=\mu gh}$,

where is the mass of the chain (mass per unit length). The arch is at the as yet unknown height above the ball and above the ground. The tension is consequently in the arch, but has decreased again immediately above the ball . ${\displaystyle \mu }$ ${\displaystyle L}$ ${\displaystyle H+L}$ ${\displaystyle F_{\parallel }(H+L)}$ ${\displaystyle F_{\parallel }(H)}$

speed

The speed results from the consideration of the radial force in the area of ​​the arc. The curvature may be uneven, even three-dimensional, ^{from 1:13} but always the curvature of both the cause and consequence of the lateral force and thus fall out: On a short elbow length with the radius of curvature , that tension reaches out not exactly opposite directions at ; is the angular deviation . This results in a force in the direction of the (local) center of curvature with the amount ${\displaystyle d}$ ${\displaystyle r}$ ${\displaystyle d\ll r}$ ${\displaystyle F_{\parallel }}$ ${\displaystyle {\frac {d}{r}}}$

${\displaystyle F_{\perp }(r)={\frac {d}{r}}F_{\parallel }}$

This force acts as a centripetal force on the mass and thus causes the curvature of the path: ${\displaystyle F_{Z}}$ ${\displaystyle m=\mu d}$

${\displaystyle F_{\perp }(r)=F_{Z}=\mu d{\frac {v^{2}}{r}}}$

where is the longitudinal speed of the chain. When releasing after the tension falls out: ${\displaystyle v}$ ${\displaystyle {\frac {d}{r}}}$

${\displaystyle F_{\parallel }=\mu v^{2}}$

This long known result says that a chain with uniform mass coating on which no other forces except a certain tension is (smooth) along any fixed space curve move can . To do this, the speed and the initial conditions must be right.

The tensile stress determined in the previous chapter from the static weight results

${\displaystyle \mu g\ (H+L)=\mu v^{2}}$

Per unit length, this is the potential energy of the chain over the height of fall and, at the same time, the mechanical work that the long end does on the short end. Half of this work comes back in the form of kinetic energy ( ), flows towards the ground and is lost there. The other half is implemented in the vessel and creates the mold effect. ${\displaystyle H+L}$ ${\displaystyle {\frac {\mu }{2}}v^{2}}$

Impulse, repulsive force and height of the fountain

The pulse per unit length is . When multiplied, this gives the impulse per unit of time, i.e. the force required to set the chain in motion: ${\displaystyle \mu v}$ ${\displaystyle v}$

${\displaystyle F_{\text{Start}}=\mu v^{2}}$

But that is precisely the tension at the top of the arch. Immediately above the ball, the tensile stress is lower by the factor and therefore insufficient. The missing force is obviously an impacting one and is applied as a fraction of the tensile stress: ${\displaystyle F_{\parallel }}$ ${\displaystyle {\frac {H}{H+L}}}$ ${\displaystyle \alpha }$

${\displaystyle {\frac {L}{H+L}}\ \mu v^{2}=\alpha \mu v^{2}.}$

Biggins and Warner measured in experiments with a ball chain . The result was the linear relationship , which means. They also showed that there is a limit of 0.5: For this, the (pull) work available (see previous chapter) would have to be converted into kinetic energy without loss. Compared to this theoretical limit, the efficiency is . ${\displaystyle L(H)}$ ${\displaystyle L\approx H/6}$ ${\displaystyle \alpha \approx 1/7}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha /0{,}5=0{,}28}$

The authors also give a simple mechanical model for the repulsive interaction that results with parameter values ​​that fit the chain used . ${\displaystyle \alpha =1/6}$

Web links

Commons : Chain Fountain  - collection of images, videos and audio files

Individual evidence

1. ↑ Steve Mold: Self siphoning beads , YouTube, February 20, 2013.
2. ↑ ^{a b} YouTube video of Royal Society Publishing: Professor Mark Warner and Dr John S. Biggins discuss Their paper published in Proceedings A .
3. ^ ^{A b} John S. Biggins, Mark Warner: Understanding the Chain Fountain . Proc. Royal Society A 470, 2014, doi : 10.1098 / rspa.2013.0689 ( arxiv.org )
4. ↑ ^{a b c} Earth Unplugged: Amazing bead chain experiment in slow motion . YouTube, June 27, 2013.
5. ^ Anoop Grewal et al .: A chain that accelerates, rather than slows, due to collisions: How compression can cause tension . American Journal of Physics, 79, 2011, p. 723, doi : 10.1119 / 1.3583481 ( online ).
6. ↑ Examiners and Moderators: Solutions of the problems and riders proposed in the Senate-House examination (Mathematics Tripos) . MacMillan, London 1854 (quoted from Biggins & Warner, 2013).