The Falling Chain Problem Benjamin Clouser and Eric Oberla December 4, 8 Abstract We present an experimental confirmation that the falling chain problem is best characterized by inelatic collisions between succesive links in the chain. Theoretical treatments which assume energy conservation as the chain falls exhibit limiting behavior that differs from those that do not. Our experiment exploits these differences to decisively show that energy is not conserved and inelastic collisions dominate. 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 to the edge of the table as possible. The chain has length l, linear mass density µ, and the amount hanging over the edge of the table is x. We wish to determine the behavior of the chain once it is released from rest and begins to fall. We will now present an overview of two methods for deriving the equations of motion.. Assuming Energy Conservation The first derivation assumes that energy is conserved as the chain falls from the table. In effect, this means that one assumes that the collisions between the chain links are elastic. We first write the potential and kinetic energies: U = du = mgdx = µg xdx = µgx () and T = mẋ = µxẋ () Now we write the Lagrangian: L = T U = µxẋ + µgx () We define the Hamiltonian: H = ẋp x L = µxẋ gµx (4) The Hamiltonian is constant in time, so: E = µxẋ gµx (5) which in turn implies that ẋ gx = C x (6) where C is a constant. Applying the boundary condition ẋ(x ) = leads to ẋ = v = g(x x x ) (7)
. Assuming Inelastic Collisions The above derivation relies on the unlikely assumption that energy is conserved in the system. Each collision between successive links in the chain is inelastic, since initially one link is moving and the other is stationary, and after the collision both links are moving at the same speed. We know from the previous derivation that the momentum of the moving part of the chain is p = µxẋ and the potential of the chain is U = µgx. With the knowledge that F = ( U x d(xẋ) dt ) and F = dp dt, we can write: = ẋ + xẍ = gx (8) Using the relation that ẍ = dv/dt = (dv/dx)(dx/dt) = v(dv/dx) we see that Integrating and applying the boundary conditions we arrive at. Observable Behavior d dx (v x ) = gx (9) ẋ = v = g (x x x ) () We wish to experimentally show that the falling chain is a non-conservative system and obeys the equations of motion derived assuming inelastic collisions. Thusly, we must determine which physical consequences 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 x is comparable to x, this turns out to unfeasible due to the limitations of our experimental apparatus. We are then led to study the motion in the limit where x x. Taking the derivative with respect to time for both equations of motion we have: N on Conservative d dt [ẋ = g(x x x )] Conservative d dt [ẋ = g(x x x )] ẋẍ = g(ẋ + x x ẋ) ẋẍ = g(ẋ + x x ẋ) ẍ = g( + x x ) ẍ = g( + x x ) In the limit x x we see that these accelerations become constant, with: ẍ = g () and ẍ = g () for the non-conservative and conservative derivations, respectively. x Figure graphs ẍ vs. x for both derivations and clearly shows the limiting behavior. Since the accelerations become constant in this limit, we see in Figure that v becomes linear with the appropriate slope in this limit as well. These are the behaviors we will seek in our experiment. Whether or not the acceleration in our data goes as g or g when x x will tell us whether or not energy is conserved.
8 7 E not conserved E conserved a = g/ a = g/ 6 Acceleration [m/s ] 5 4 4 5 6 x/x Figure : Acceleration as a function of x/x for both derivations. 5 E not conserved, x =. E conserved, x =. 4 v [m /s ]....4.5.6 Length of Chain off Table, x [m] Figure : Velocity squared as a function of x for both derivations. x is the length of chain over the edge of the table. 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. A metal linked chain (m = 7. g, l = 64.5 cm) was coiled on the edge of a table and a length x was allowed to drop over the edge. A string was tied to the link that was immediately over the table edge and strung over a pulley with a light counterweight (m =.8 g) on the other side. This counterweight keeps the string taut as it passes over the pulley, which is also a rotary encoder and allows us to take position, velocity and acceleration data with a computer. We used the Logger Pro software to take the data. The rotary encoder consists of 5 spokes, which block and
unblock LEDs located inside the encoder. Having LEDs instead of just one allows the software to assign a direction to the displacement it measures. Figure : Experimental setup. The figure shows the string looped over the rotary encoder and attached to the counterweight on one end and the chain at the other. We tested this set-up by varying x and the configuration of the chain on the table. Two configurations were tried: ) s-coil, where the chain was coiled in a repeated s shape on the table edge and ) piled, where the chain was piled on top of itself on the table edge. Piling the chain seems to result in very uneven uncoiling, which is noticeable in the data. Putting the chain in an s-coil is more repeatable and results in more regular data. One problem with this method is that it is not valid when x x. When this is the case, the length of chain x 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 x portion of the chain, while the rest of the chain continues to fall quickly to the ground. In effect, we end up measuring only the fact that the x 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 the chain and the table edge. This was not taken into account by our theory, so we wanted to try an experiment that eliminated this issue. Our second method is shown in Figure 4. In this set-up, we used a beaded chain (see previous figure) that was draped over the pulley located at a height, h = 8.8 cm, above the table. The beaded chain had mass, m = 7.4 g, streched length, ls = 94.5 cm and compressed length lc = 7.8 cm. Although this setup seemed to avoid the problems of the chain interacting with the table s edge, it still does not correctly model the problem in the x x limit. We had great difficulty keeping the beaded chain running over the pulley during the entirety of its fall. Any irregularites in the motion as the chain unpiled itself were enough for the chain to jump the tracks and foul up the data. In order to combat this we constructed a channel of sorts out of two index cards. These cards were attached to either side of the rotary encoder s pulley and did help to keep the chain in place. In reality, though, the channel did not address the root issue of the chain bouncing off the pulley, it only ensured that the chain landed again on the pulley after it bounced off. 4
Figure 4: Experimental setup. The beaded chain is looped over the rotary encoder, which spins and directly records its movement. Call h the distance from the table up to the pulley, and x the amount of chain hanging from the top of the pulley. A mathematical treatment of this problem shows that it results in the same limiting behavior as the original setup. The cut up index cards on the rotary encoder served 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 Second Law becomes: d [(x + h)x ] = µg(x h) () dt After some manipulation, integration, and application of initial conditions, equation () becomes: µ (x + h) x = g(x x )(x + x x + x h ) (4) Examining this equation leads us to conclude that if x = h, then the acceleration is always constant in time with a magnitude of g. Although this behavior is interesting, and our data did reflect it to some extent, it is clearly not contained in either of the equations derived to describe the standard falling chain problem. That said, this configuration still tends towards x = g in the x x limit. Data Figure 5 below shows velocity data taken by our first experimental method with four different initial conditions. We took data with x =.5 cm and cm for both s-coil and pile configurations of the chain. We concluded both from observing the chain fall as well as the presented data that the s-coil method limited unwanted friction of the uncoiling chain. Careful inspection of the velocity data for the chain initially in a pile shows oscillatory behavior. We surmise that this behavior is due to the chain unpiling in an uneven manner. Accordingly, data from the s-coil method will be used to compare with theoretical predictions. Wong, C. H., S. H. Youn, K. Yasui, The falling chain of Hopkins, Tait, Steele, and Cayley. 5
6 4 x =.5 cm, pile x = cm, pile x =.5 cm, s-coil x = cm, s-coil Velocity [cm/s] 8 6 4 5 5 5 5 4 Length of Chain off Table [cm] Figure 5: 4 data sets from our first experimental configuration. These data compare piled and s-coiled chain configurations for x =.5 cm and cm. x =. cm x = 5. cm x =. cm x =. cm 8 Velocity [cm/s] 6 4 5 5 5 5 4 Length of Chain off Table [cm] Figure 6: Data sets for our second experimental setup. Note that the x values quoted on the graph are actually x h. Data were taken with x h =. cm, 5. cm,. cm, and. cm. This setup exhibits the oscillatory beat behavoir much more strongly than the first setup. Once again, we believe these beats to be due to the chain unpiling unevenly. This claim is supported by the fact that the beats seem to occur over a characteristic distance of 7 8 cm, which is roughly twice the diameter of the chain pile. 4 Results Results from our experiment can be seen in the next figures. We first consider the findings from experimental setup. 6
5 5 v [cm /s ] 5 x = 5 cm x = cm fit to x =5 cm, v(x) = 7.x - 4 fit to x = cm, v(x) = 76.7x + 7 5 5 5 5 4 Length of Chain off Table, x [cm] Figure 7: Plot of v vs. x, the length of chain that has fallen off the table. When energy is not conserved, equation () shows that v (x) approaches a linear function with slope g as x x. A linear fit was performed in this region. Our results showed a slope of 7. m s for x = 5 cm and 7.6 m s for x = cm. The expected result, 6.5 m s, is somewhat consistent with our data. It makes sense that value obtained for x = 5 cm is closer to the expected value than that for x = cm since the limit x x is more easily attained. 6 4 x = 5 cm x = cm fit (x = 5 cm) v(t) =.9t -.4 fit (x = cm) v(t) = 7.t + 7. Velocity [cm/s] 8 6 4....4.5.6.7.8 Time [s] Figure 8: Plot of v vs. t. In our analysis, we apply our previous discussion of the limiting behavior of the acceleration. 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. m s for x = 5 cm and.7 m s for x = cm. The expected value, as shown in equation (), is g or.7 m s for inelastic collisions. Our results confirm this. 7
Velocity [m/s].5 x =.5 cm, s-coil Theory, x =.5 cm x = cm, s-coil Theory, x = cm.5....4.5.6.7.8 Length of Chain off Table [m] Figure 9: Comparison of theoretical data with experimental data in a plot of v(x). Our data shows the same shape and asymptotic behavior as the theory, but is consistently less by what appears to be a constant offset. This is most likely the result of some systematic friction and is the reason why we wanted to focus on the limiting region in the first place. 8 Velocity [cm/s] 6 4 x =. cm x =. cm x =. cm fit (x =. cm) v(t)=86.t-67.4 fit (x =. cm) v(t)=8.t-9.7 fit (x =. cm) v(t)=8.t-5.4..4.6.8 Time [s] Figure : Results from experimental setup. We plotted v vs. t and fitted the data with a line in the asymptotic region. As with the first set-up, we expect a value of a = g in this region, but our data suggests something times less. We conclude that our first set up was more consistent. By attempting to eliminate interactions between the table edge and the chain, we appear to have introduced more 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 suggests that it will reach a constant acceleration once a sufficient length has fallen off the table. This acceleration 8
depends on whether energy is taken to be conserved (a = g ) or not (a = g ) 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 the linear region, we found a constant acceleration that was close to g. A comprehensive error analysis was not 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 not conserved. There was a clear discrepancy between the data from the two setups and several factors are likely to blame. In setup, we used a beaded chain that was allowed to fall over a pulley, where the rotation of the pulley gave us our data. Since our data gave results less than expected, it is possible that the chain slid on the pulley and some motion data was lost (ie v > ωr). By placing a counterweighted string over the pulley as in setup, it was more likely that v = ωr and the rotation of the pulley gave an accurate description of the motion of the chain. Our results, however surprising, seem to confirm the treatment of the falling chain as a series of inelastic collisions. The experimental setups were somewhat crude, but still gave reasonable and repeatable data. Our first experimental setup was subjected to different initial conditions and considerable disturbances yet still managed to yield data that was consistent in the asymptotic, suggesting that behavior in the limit x x is fairly robust. 6 Acknowledgements We would like to express our gratitude to Van Bistrow for graciously allowing us to use University equipment and laboratory space. 9