Can someone explain this solution for the motion of a falling chain?

In an example of Marion's classical dynamics 5th edition, I found example 9.2 not making sense, which states:    My questions are:

1. The horizontal motion cannot be ignored even in the idealized case. As the velocity $\dot{x}$ tends to infinity, the horizontal momentum (in the direction of $y$) must grow to a certain value as $x \rightarrow b$, which implies $\dot{x} \rightarrow \infty$ (and in reality, $x$ and $y$ direction motion are kind of dependent on each other). In other words, if the horizontal motion will always be observed, then how can he assume no horizontal motion? Also since $\dot{x} \rightarrow \infty$, how can he not use theory of relativity?

2. Energy is not conserved, since there is an additional tension force on the right side and doing work, compared to free fall. (In other words, the tension is contributing a certain amount of energy to $K$)

3. After it comes to rest, where the heck does the energy go?

4. An experiment done indicates that the tension force at A is 25 times that of the weight. Therefore I can't help wondering how much difference of order of magnitude can be treated as "infinity"? (order of one certainly not...)

1. If the chain is folded up on itself with the initial horizontal separation then yes it can be ignored. The picture is misleading because it shows a gentle bend connecting the two sides when in idealized reality you have a discontinious sharp bend at the bottom. Remember ideally there is not flexular rigidity (no resistance to bending) and inflexibiliy (no potential energy stored due to tension).

2. Tension is an internal force and does no work.

3. In an ideal case it does not come to rest. It flips between potential and kinetic energy. As soon as full extension is reached there is an impulse developing from the support reversing all the vertical velocities at the same time. The chain the proceeds to fold itself up until it reaches $x=0$ when the velocities all becomes zero and the process starts again. This is like the snapping of a whip which reverses direction as soon as it is fully extended.

4. Infinity is the result of equation of motion in the ideal case. In reality the extensibility of the chain, the speed of sound along the chain and the flexibility of the support contribute effects that become dominant near full extension $x=b$ with the effect of whipping the chain which either breaks, or bounces back as explained above, but with reduced energy. I really do not understand you question about how much is infinity. An experiment is never going to show infinity and what the test really shows is a measure of the flexibility of all the parts.

• 2.) If tension is an internal force, then how can tension affect the motion of right side? why tension did make a difference from free fall? – Shing Apr 17 '15 at 14:22
• 1.) taking y is equal to zero, but as x approach to b, $\dot{x}$ approaches to infinity, means $\dot{y}$ approaches to infinity. I don't think "infinity times zero" can be ignored. – Shing Apr 17 '15 at 14:24
• 4.) I mean if a theory predict infinity. then in reality (experiment) by how many order of magnitude can be thought as "infinity"? – Shing Apr 17 '15 at 14:25
• 1) If the chain is falling on the right side, after it whips it is going to be rising on the left side. That horizontal motion is going to carry kinetic energy but only for a negligible fraction of time. After that is going to loose all horizontal motion. So when looking at before and after only the vertical motion accounts for change in kinetic energy. – ja72 Apr 17 '15 at 14:30
• A good answer to get you to 10k. +1 to you! – Floris Apr 17 '15 at 22:43

The kinetic energy of the segments which come to rest is transferred to the moving part of the chain. This is the non-intuitive part of the experiment. Here is a video from Veritasium chanel, which explains it.

For real chain as soon as the last segment reaches the bottom the horizontal motion is no longer neglectable. The last segment of the chain starts rotating and there is no next segment to which it could transfer energy so it starts swinging. The swinging causes the whole chain to wobble. The enegry of this wobbling motion then dissipates. Please keep in mind that the example is only aproximation. The aproximation is good as long as there are many segments on the right side.

I took $b=10m$ and length of segment $\Delta x=5cm$. When the last segment reaches the bottom of the chain it has traveled distance $$x=b-\Delta x=9.95m$$ When I plug in equation 9.17 I get the speed of the last segment of the chain: $$(\dot x)^2=19620(ms^{-1})^2$$ $$\underline{\dot x=140ms^{-1}}$$ It's pretty high but certainly not relativistic.

The effect is important for function of whip. Many people think that the sound of whip is caused by the whip slaping itself. The reality is much more interesting. The end of the whip actually reaches supersonic speed!