The Falling Chain Problem

Benjamin Clouser and Eric Oberla

December 4, 2008

Abstract

We present an experimental confirmation that the falling chain problem is best characterized byinelatic collisions between succesive links in the chain. Theoretical treatments which assume energyconservation as the chain falls exhibit limiting behavior that differs from those that do not. Ourexperiment exploits these differences to decisively show that energy is not conserved and inelasticcollisions dominate.

1 Theoretical Motivations

A length of chain hangs at rest over the edge of a table, and the rest of the chain is bunched up as close tothe edge of the table as possible. The chain has length l, linear mass density µ, and the amount hangingover the edge of the table is x0. We wish to determine the behavior of the chain once it is released fromrest and begins to fall. We will now present an overview of two methods for deriving the equations ofmotion.

1.1 Assuming Energy Conservation

The first derivation assumes that energy is conserved as the chain falls from the table. In effect, thismeans that one assumes that the collisions between the chain links are elastic. We first write the poten-tial and kinetic energies:

U =∫dU = −

∫mgdx = −µg

∫xdx = −1

2µgx2 (1)

andT =

12mx2 =

12µxx2 (2)

Now we write the Lagrangian:

L = T − U =12µxx2 +

12µgx2 (3)

We define the Hamiltonian:H = xpx − L =

12µxx− 1

2gµx2 (4)

The Hamiltonian is constant in time, so:

E =12µxx− 1

2gµx2 (5)

which in turn implies that

x2 − gx =C

x(6)

where C is a constant. Applying the boundary condition x(x0) = 0 leads to

x2 = v2 = g(x− x20

x) (7)

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1.2 Assuming Inelastic Collisions

The above derivation relies on the unlikely assumption that energy is conserved in the system. Eachcollision between successive links in the chain is inelastic, since initially one link is moving and the otheris stationary, and after the collision both links are moving at the same speed. We know from the previousderivation that the momentum of the moving part of the chain is p = µxx and the potential of the chainis U = − 1

2µgx2. With the knowledge that F = −(∂U

∂x ) and F = dpdt , we can write:

d(xx)dt

= x2 + xx = gx (8)

Using the relation that x = dv/dt = (dv/dx)(dx/dt) = v(dv/dx) we see that

12d

dx(v2x2) = gx2 (9)

Integrating and applying the boundary conditions we arrive at

x2 = v2 =2g3

(x− x30

x2) (10)

1.3 Observable Behavior

We wish to experimentally show that the falling chain is a non-conservative system and obeys theequations of motion derived assuming inelastic collisions. Thusly, we must determine which physicalconsequences of the equations are in starkest contrast in order to best determine which regime dominates.Although it is tempting to study the motion in the limit where x0 is comparable to x, this turns out tounfeasible due to the limitations of our experimental apparatus. We are then led to study the motionin the limit where x � x0. Taking the derivative with respect to time for both equations of motion wehave:

Non− Conservative Conservative

d

dt[x2 =

23g(x− x3

0

x2)]

d

dt[x2 = g(x− x2

0

x)]

2xx =23g(x+ 2

x30

x2x) 2xx = g(x+

x20

x2x)

x =13g(1 +

x30

x3) x =

12g(1 +

x20

2x2)

In the limit x� x0 we see that these accelerations become constant, with:

x =13g (11)

andx =

12g (12)

for the non-conservative and conservative derivations, respectively.Figure 1 graphs x vs. x

x0for both derivations and clearly shows the limiting behavior. Since the

accelerations become constant in this limit, we see in Figure 2 that v2 becomes linear with the appropriateslope in this limit as well. These are the behaviors we will seek in our experiment. Whether or not theacceleration in our data goes as g

3 or g2 when x� x0 will tell us whether or not energy is conserved.

2

0

1

2

3

4

5

6

7

8

1 2 3 4 5 6

Acc

eler

atio

n [m

/s2 ]

x/x0

E not conservedE conserved

a = g/3a = g/2

Figure 1: Acceleration as a function of x/x0 for both derivations.

0

1

2

3

4

5

0.1 0.2 0.3 0.4 0.5 0.6

v2 [m2 /s

2 ]

Length of Chain off Table, x [m]

E not conserved, x0 = 0.1E conserved, x0 = 0.1

Figure 2: Velocity squared as a function of x for both derivations. x is the length of chain over the edgeof the table.

2 Experimental Setup

We performed two experiments to test energy conservation of the falling chain. For the first experiment,we used the set-up presented in Figure 3. A metal linked chain (m = 37.2 g, l = 64.5 cm) was coiled onthe edge of a table and a length x0 was allowed to drop over the edge. A string was tied to the link thatwas immediately over the table edge and strung over a pulley with a light counterweight (m = 2.8 g)on the other side. This counterweight keeps the string taut as it passes over the pulley, which is also arotary encoder and allows us to take position, velocity and acceleration data with a computer. We usedthe Logger Pro software to take the data. The rotary encoder consists of 512 ‘spokes’, which block and

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unblock 2 LEDs located inside the encoder. Having 2 LEDs instead of just one allows the software toassign a direction to the displacement it measures.

Figure 3: Experimental setup 1. The figure shows the string looped over the rotary encoder and attachedto the counterweight on one end and the chain at the other.

We tested this set-up by varying x0 and the configuration of the chain on the table. Two configurationswere tried: 1) ‘s-coil’, where the chain was coiled in a repeated ‘s’ shape on the table edge and 2) piled,where the chain was piled on top of itself on the table edge. Piling the chain seems to result in veryuneven uncoiling, which is noticeable in the data. Putting the chain in an ‘s-coil’ is more repeatable andresults in more regular data.

One problem with this method is that it is not valid when x ≈ x0. When this is the case, the lengthof chain x0 hanging over the table has a mass which is fairly close to the mass of the counterweight.When the chain starts falling, the counterweight holds up the x0 portion of the chain, while the rest ofthe chain continues to fall quickly to the ground. In effect, we end up measuring only the fact that thex0 portion of the chain weighs about as much as the counterweight.

Another problem with the first experimental set-up seemed to be the large interaction between thechain and the table edge. This was not taken into account by our theory, so we wanted to try anexperiment that eliminated this issue. Our second method is shown in Figure 4. In this set-up, we useda beaded chain (see previous figure) that was draped over the pulley located at a height, h = 18.8 cm,above the table. The beaded chain had mass, m = 37.4 g, streched length, ls = 94.5 cm and compressedlength lc = 71.8 cm.

Although this setup seemed to avoid the problems of the chain interacting with the table’s edge, itstill does not correctly model the problem in the x ≈ x0 limit. We had great difficulty keeping the beadedchain running over the pulley during the entirety of its fall. Any irregularites in the motion as the chainunpiled itself were enough for the chain to jump the tracks and foul up the data. In order to combat thiswe constructed a channel of sorts out of two index cards. These cards were attached to either side of therotary encoder’s pulley and did help to keep the chain in place. In reality, though, the channel did notaddress the root issue of the chain bouncing off the pulley, it only ensured that the chain landed againon the pulley after it bounced off.

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Figure 4: Experimental setup 2. The beaded chain is looped over the rotary encoder, which spins anddirectly records its movement. Call h the distance from the table up to the pulley, and x0 the amountof chain hanging from the top of the pulley. A mathematical treatment of this problem shows that itresults in the same limiting behavior as the original setup. The cut up index cards on the rotary encoderserved as channels to keep the chain on the pulley.

The theory behind this experiment is slightly different than the previous set-up, as Newton’s SecondLaw becomes:

µd

dt[(x+ h)x] = µg(x− h) (13)

After some manipulation, integration, and application of initial conditions, equation (13) becomes:

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(x+ h)2x2 =13g(x− x0)(x2 + x0x+ x2

0 − 3h2) (14)

Examining this equation leads us to conclude that if x0 = 2h, then the acceleration is always constantin time with a magnitude of g

3 . Although this behavior is interesting, and our data did reflect it to someextent, it is clearly not contained in either of the equations derived to describe the standard falling chainproblem. That said, this configuration still tends towards x = 1

3g in the x� x0 limit.1

3 Data

Figure 5 below shows velocity data taken by our first experimental method with four different initialconditions. We took data with x0 = 10.5 cm and 30 cm for both ‘s-coil’ and pile configurations of thechain. We concluded both from observing the chain fall as well as the presented data that the ‘s-coilmethod limited unwanted friction of the uncoiling chain. Careful inspection of the velocity data for thechain initially in a pile shows oscillatory behavior. We surmise that this behavior is due to the chainunpiling in an uneven manner. Accordingly, data from the ‘s-coil’ method will be used to compare withtheoretical predictions.

1Wong, C. H., S. H. Youn, K. Yasui, The falling chain of Hopkins, Tait, Steele, and Cayley.

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0

20

40

60

80

100

120

140

160

0 5 10 15 20 25 30 35 40

Vel

ocity

[cm

/s]

Length of Chain off Table [cm]

x0 = 10.5 cm, pilex0 = 30 cm, pile

x0 = 10.5 cm, s-coilx0 = 30 cm, s-coil

Figure 5: 4 data sets from our first experimental configuration. These data compare piled and s-coiledchain configurations for x0 = 10.5 cm and 30 cm.

0

20

40

60

80

100

0 5 10 15 20 25 30 35 40

Vel

ocity

[cm

/s]

Length of Chain off Table [cm]

x0 = 3.2 cmx0 = 5.2 cm

x0 = 11.2 cmx0 = 21.2 cm

Figure 6: Data sets for our second experimental setup. Note that the x0 values quoted on the graphare actually x0 − h. Data were taken with x0 − h = 3.2 cm, 5.2 cm, 11.2 cm, and 21.2 cm. This setupexhibits the oscillatory ‘beat’ behavoir much more strongly than the first setup. Once again, we believethese ‘beats’ to be due to the chain unpiling unevenly. This claim is supported by the fact that the beatsseem to occur over a characteristic distance of 7−8 cm, which is roughly twice the diameter of the chainpile.

4 Results

Results from our experiment can be seen in the next figures. We first consider the findings from experi-mental setup 1.

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0

5000

10000

15000

20000

25000

0 5 10 15 20 25 30 35 40

v2 [cm

2 /s2 ]

Length of Chain off Table, x [cm]

x0 = 5 cmx0 = 30 cm

fit to x0=5 cm, v(x) = 711.1x - 4012fit to x0=30 cm, v(x) = 760.7x + 2172

Figure 7: Plot of v2 vs. x, the length of chain that has fallen off the table. When energy is not conserved,equation (10) shows that v2(x) approaches a linear function with slope 2

3g as x � x0. A linear fit wasperformed in this region. Our results showed a slope of 7.11 m

s2 for x0 = 5 cm and 7.61 ms2 for x0 = 30 cm.

The expected result, 6.53 ms2 , is somewhat consistent with our data. It makes sense that value obtained

for x0 = 5 cm is closer to the expected value than that for x0 = 30 cm since the limit x0x → 0 is more

easily attained.

0

20

40

60

80

100

120

140

160

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Vel

ocity

[cm

/s]

Time [s]

x0 = 5 cmx0 = 30 cm

fit (x0= 5 cm) v(t) = 332.9t - 102.4fit (x0= 30 cm) v(t) = 372.1t + 27.2

Figure 8: Plot of v vs. t. In our analysis, we apply our previous discussion of the limiting behavior of theacceleration. Our data is fit with a line in the asymptotic region and the results are shown on the plot.The results of the fit gave a slope of 3.32 m

s2 for x0 = 5 cm and 3.72 ms2 for x0 = 30 cm. The expected

value, as shown in equation (11), is g3 or 3.27 m

s2 for inelastic collisions. Our results confirm this.

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0

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Vel

ocity

[m/s

]

Length of Chain off Table [m]

x0 = 10.5 cm, s-coilTheory, x0 = 10.5 cm

x0 = 30 cm, s-coilTheory, x0 = 30 cm

Figure 9: Comparison of theoretical data with experimental data in a plot of v(x). Our data showsthe same shape and asymptotic behavior as the theory, but is consistently less by what appears to bea constant offset. This is most likely the result of some systematic friction and is the reason why wewanted to focus on the limiting region in the first place.

0

20

40

60

80

100

0 0.2 0.4 0.6 0.8 1

Vel

ocity

[cm

/s]

Time [s]

x0 = 3.2 cmx0 = 11.2 cmx0 = 21.2 cm

fit (x0=3.2 cm) v(t)=186.0t-67.4fit (x0=11.2 cm) v(t)=208.0t-29.7fit (x0=21.2 cm) v(t)=238.1t-15.4

Figure 10: Results from experimental setup 2. We plotted v vs. t and fitted the data with a line inthe asymptotic region. As with the first set-up, we expect a value of a = g

3 in this region, but ourdata suggests something ∼ 2 times less. We conclude that our first set up was more consistent. Byattempting to eliminate interactions between the table edge and the chain, we appear to have introducedmore friction by dragging the beaded chain over the pulley.

5 Conclusion

We have presented our solution to the falling chain problem. The theory behind the falling chain suggeststhat it will reach a constant acceleration once a sufficient length has fallen off the table. This acceleration

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depends on whether energy is taken to be conserved (a = g2 ) or not (a = g

3 ) in the equations of motion.After trying several experimental setups, we settled for the two presented here. The data from our

first experimental setup gave us more consistent results. Upon fitting a v vs. t plot of our data in thelinear region, we found a constant acceleration that was close to g

3 . A comprehensive error analysis wasnot performed in this experiment so it is difficult to determine the specific range of our uncertainty.However, our data was repeatable and it is realistic to conclude our experiment showed energy is notconserved.

There was a clear discrepancy between the data from the two setups and several factors are likely toblame. In setup 2, we used a beaded chain that was allowed to fall over a pulley, where the rotation ofthe pulley gave us our data. Since our data gave results less than expected, it is possible that the chainslid on the pulley and some motion data was lost (ie v > ωr). By placing a counterweighted string overthe pulley as in setup 1, it was more likely that v = ωr and the rotation of the pulley gave an accuratedescription of the motion of the chain.

Our results, however surprising, seem to confirm the treatment of the falling chain as a series of in-elastic collisions. The experimental setups were somewhat crude, but still gave reasonable and repeatabledata. Our first experimental setup was subjected to different initial conditions and considerable distur-bances yet still managed to yield data that was consistent in the asymptotic, suggesting that behaviorin the limit x� x0 is fairly robust.

6 Acknowledgements

We would like to express our gratitude to Van Bistrow for graciously allowing us to use Universityequipment and laboratory space.

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