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The many variants of falling chain problems that have been published in the physics literature over the last two centuries are here analyzed using generalized Euler–Lagrange equations that include momentum flux. Familiar results are recovered, and new ones are presented. We demonstrate and clarify the role played by momentum flux by considering the falling chains as both variable-mass and fixed-mass systems. We calculate the evolution of energy as a useful parameter that provides insight into some of the unexpected behavior of falling chain systems. Our Lagrangian approach incorporates all published falling chain phenomena (including the whiplash effect) within a unifying framework, highlighting connections between the phenomena and providing a valuable pedagogical example of variable mass systems analysis.The many variants of falling chain problems that have been published in the physics literature over the last two centuries are here analyzed using generalized Euler–Lagrange equations that include momentum flux. Familiar results are recovered, and new ones are presented. We demonstrate and clarify the role played by momentum flux by considering the falling chains as both variable-mass and fixed-mass systems. We calculate the evolution of energy as a useful parameter that provides insight into some of the unexpected behavior of falling chain systems. Our Lagrangian approach incorporates all published falling chain phenomena (including the whiplash effect) within a unifying framework, highlighting connections between the phenomena and providing a valuable pedagogical example of variable mass systems analysis.

Magnets are attached to a falling object, which is allowed to fall through a conducting ring. A magnetic impulsive force produced by the ring reduces (very slightly) the velocity of the object. The impulsive force is fully characterized using a novel method for measuring the velocity and acceleration of the object, which allows for determination of the gravitational field strength, to within 0.2% of the published local value. The experimentally obtained impulsive force is well explained by the calculated Lorentz force acting on the ring.

We consider the uniform motion of a short strong cylindrical magnet falling inside a conducting pipe and study the dependence of the magnetic braking force on the distance of the falling magnet from the pipe wall. We also consider two magnets falling together with parallel or opposite magnetic moments. We develop models for these three cases of magnetic braking and describe the experiments that validate our models. The experimental setups are inexpensive and can be readily assembled in a teaching laboratory.

A chain is firmly attached at one end while the other falls freely in the gravitational field. We report careful time-resolved measurements of the horizontal and vertical components of the force applied by the chain on the mount. Our results complement previous laboratory measurements and simulations of the free end dynamics.

We analyze the initial motion of an elastic bar that is suddenly released after being hung from one end. The analytical solutions uncover some unexpected properties, which can be checked with a digital camera or camcorder in an alternative setup in which a spring is substituted for the bar. The model and the experiments are useful for understanding the similarities and differences between the elastic properties of bars and springs. Students can use the simple experiments to improve their understanding of elastic waves.

A simple setup was assembled to study the motion of an object while it falls. The setup was used to determine the instantaneous velocity, terminal velocity and acceleration due to gravity. Also, since the whole project was done within $20 it can easily be popularized.

There has been great interest in applying the results of statistical mechanics to single molecule experiements. Recent work has highlighted so-called non-equilibrium work-energy relations and Fluctuation Theorems which take on an equilibrium-like (time independent) form. Here I give a very simple heuristic example where an equilibrium result (the barometric law for colloidal particles) arises from theory describing the {\em thermodynamically} non-equilibrium phenomenon of a single colloidal particle falling through solution due to gravity. This simple result arises from the fact that the particle, even while falling, is in {\em mechanical} equilibrium (gravitational force equal the viscous drag force) at every instant. The results are generalized by appeal to the central limit theorem. The resulting time independent equations that hold for thermodynamically non-equilibrium (and even non-stationary) processes offer great possibilities for rapid determination of thermodynamic parameters from single molecule experiments., Comment: 6 pages

A theoretical analysis about the influence of the Hertz elastic contact on a three body chain collision is presented. In spite of the elastic character of the collision, the final velocity of each particle depends on the particular interaction between them. A system involving two elastic spheres falling together one in top of the other under the action of gravity, and colliding with an horizontal hard wall is studied in detail. The effect of the Hertz contact interaction can be easily put in evidence for some particular situations., 5 pages, 6 figures

Dimensional analysis is a simple, physically transparent and intuitive method for obtaining approximate solutions to physics problems, especially in mechanics. It may—indeed sometimes should—precede or even supplant mathematical analysis. And yet dimensional analysis usually is given short shrift in physics textbooks, presented mostly as a diagnostic tool for finding errors in solutions rather than in finding solutions in the first place. Dimensional analysis is especially well suited to estimating the magnitude of errors associated with the inevitable simplifying assumptions in physics problems. For example, dimensional arguments quickly yield estimates for the errors in the simple expression 2h/g for the descent time of a body dropped from a height h on a spherical, rotating planet with an atmosphere as a consequence of ignoring the variation of the acceleration due to gravity g with height, rotation, relativity, and atmospheric drag.

The problem of free fall (rising or falling) of an object near the surface of the rotating earth is revisited. Expressions for the eastward and westward deflections of the object, which are normally explained by Coriolis force, are derived directly from conservation of angular momentum of the object with respect to the center of the earth. Launching conditions for zero lateral (east–west) deflections are also discussed. The treatment is simple enough, both conceptually and mathematically, to be presented in an introductory physics course.

We describe an experiment to determine the acceleration due to gravity, g, with an accuracy of about 1 part in 104. The experiment was designed to expose students to critical thinking in collecting, selecting, and analyzing data, and interpreting the results.

We revisit a recent analysis of the time of fall of a magnet as it slows down while passing through a conducting tube. We complement recent work by considering the effect of the thickness of the tube on the time of fall. The resulting expression gives a more accurate expression for the time of fall.

The Bohm criterion is an inequality signifying that the ion flow speed at the plasma boundary must be at least as great as the ion sound speed in order for a sheath to form at the boundary. A physical explanation for this phenomenon is given, and the phenomenon is compared with the flow of falling water.

The problem of an object falling from high altitudes where the variation of atmospheric pressure cannot be neglected is investigated. The equation of motion for the variation of the velocity of the object as a function of altitude is solved exactly. The results show that, unlike an object falling in a uniform atmosphere whose speed monotonically increases and approaches the terminal speed, the speed of a high‐altitude falling object first increases, goes through a maximum, and then decreases and approaches the terminal speed from above. The results also show that if the initial altitude of the object is greater than a critical value, the object always strikes the ground with a speed that is higher than its terminal speed by a finite value, in contrast to the case of a freely falling object in a uniform atmosphere.

The motion of a spring, initially suspended from its top end and hanging in equilibrium in the earth’s field, and then released and allowed to fall, is studied. The difference between the behaviors of loosely wound and tightly wound springs is emphasized.

Fall streaks are arcs in the sky composed of ice crystals falling from high‐altitude cirrus clouds. With the crystals falling at their vertical terminal velocity and their horizontal velocity that of the local wind, the streaks are parabolas if the wind increases linearly with height. The dynamics of fall streaks is formally identical with that of drag‐free projectiles−but with a twist: the roles of horizontal and vertical coordinates are reversed.

A simple strategy is presented for solving the “inverse rocket” problem of a particle accumulating material from a medium through which it falls vertically. Some forms of drag can also be easily included, thereby changing the constant acceleration to a more realistic value.

‘‘A chain is suspended, both ends at the same elevation, and then one end is released.’’ This familiar problem is studied and the real dynamical behavior is found to be very different from the textbook answer. Results are presented that show that the real chain is largely an energy‐conserving system. This gives rise to acceleration of the free end greater than g, and to very large tension at the fixed end.

Spark‐recorded speed and distance data for a freely falling body are analyzed by the method of least squares to obtain the acceleration due to gravity g. It is found that the resistance and buoyancy of the air produce systematic errors that are not negligible. These factors are taken into account both by an approximate correction and by an exact nonlinear least‐squares analysis of the data. From a wind tunnel measurement the drag coefficient of the falling body is found to be CD = 0.63±0.03. Then from the distance data for ten trials we obtain g = 9.8012 ±0.0035 m/sec2. This value agrees with the nominal value of g for our location, 9.8026 m/sec2, within one standard deviation.

The initial acceleration of a rigid rod rotating around one end, and the time required for the rod to fall onto a horizontal table, is compared with the acceleration and falling time of a free particle originally placed on the rod. The conditions under which the rod falls faster than the particle are established. Three cases are identified. The third case, in which the particle catches up with the rod, has not, to the author’s knowledge, been published previously.

Effects producing the semidiurnal tidal bulges are described, using elementary methods, without reference to fictitious forces and without recourse to a potential. One analysis presents a rough but simple analog to the tidal effects in terms of gedanken experiments in a very large, freely falling space laboratory. Another analysis explicitly contrasts the effects that would be observed on the earth in a ’’one‐body’’ system (i.e., earth held fixed in an inertial frame; the moon revolving around it) with effects in a ’’two‐body’’ system (i.e., earth free; earth‐moon system revolving in the inertial frame).

The problem of a cylinder falling under the action of gravity in contact with a larger fixed cylinder is discussed from a qualitative point of view, both theoretically and experimentally. The same approach can be followed in a number of problems, and we think, can lead to a better understanding of classical mechanics, at least for the beginning student.

When height of fall is plotted against frequency of coiling in the liquid rope-coil experiment a straight line relationship that does not pass through the origin results. A correction due to the effect of surface tension on the stream of falling liquid yields “effective” heights of fall which, when plotted against the rate of coiling, does result in a straight line through the origin.

The tendency for a falling stream of very viscous liquid to coil like a rope when it strikes a horizontal surface is studied and the building of a “cone” of liquid of considerable height above the surface is observed. It is shown that, although the liquid stream as a whole appears to rotate about the axis of the “cone,” the particles of liquid do not actually rotate about this axis. Variations of the liquid rope-coil effect are discussed.

The connection between an accelerated charge and one at rest in a (weak) gravitational field is discussed in accordance with the principle of equivalence. For that purpose, the fields produced by a freely falling charge and a supported one (i.e., at rest in a gravitational field) are transformed to the rest-frame of the observer, who may be similarly supported or freely falling. A nonvanishing energy flux is found only if the charge is freely falling and the observer supported, or vice versa. This agrees with previously established results.

Particles falling in a fluid rotating about a horizontal axis can go into quasistable orbits. An analytical solution of the particle’s motion is derived. This model may be applicable to suspension of diatoms and other particles in the ocean.

It is shown that investigating Coriolis effects in the fixed frame leads to a clearer understanding of their physical nature: the body carries along its initial velocity, larger or smaller than the velocity of its corresponding point during the motion. A simple but thorough calculation of some effects (the Foucault pendulum and the eastward drift of a falling body) is given.

We have performed a straightforward vertical fall experiment for a case where the effects of air resistance are important and directly measurable. Using a commonly available badminton shuttlecock, a tape measure, and a millisecond timer, the times required for the shuttlecock to fall given distances (up to almost ten meters) were accurately measured. The experiment was performed in an open stairwell. The experimental data was compared to the predictions of several models. The best fit was obtained with the model which assumes a resistive force quadratic in the instantaneous speed of the falling object. This model was fitted to the experimental data enabling us to predict the terminal velocity of the shuttlecock (6.80 m/sec). The results indicate that, starting from rest, the vertically falling shuttlecock achieves 99% of its terminal velocity in 1.84 sec, after falling 9.2 m. The relative ease in collecting the data, as well as the excellent agreement with theory, make this an ideal experiment for use in physics courses at a variety of levels.

An analog electronic simulator for the motion of an object projected horizontally in a vertical acceleration field is described. The device features include adjustable values for both the horizontal speed and the strength of ’’gravity.’’ A metronome circuit gives aural and graphic time information. The trajectories are displayed on a storage oscilloscope. Photographs of typical families of curves are shown. [The apparatus described in this article was first prize winner in the College Demonstration category of the 1975 AAPT Apparatus Competition—Editor]

When a tall isolated chimney or smokestack is destroyed, usually a charge is detonated at its base and the chimney falls in the way a tree falls in that the base remains fixed. The typical photograph of the chimney falling shows it breaking as it falls at a point about one‐third its length from the bottom. Application of Newton’s laws of motion to the problem of a long thin chimney of constant mass per unit length shows that there is a maximum bending torque exerted by the upper part of the chimney on the lower when the lower part consists of the lower one‐third of the chimney.

A didactical apparatus for the absolute measurement of small coefficients of viscosity (1 centipoise) with an accuracy of 1%is presented. A W-shaped piece rotates about a vertical axis at a variable distance from the walls of an identically shaped vessel, containing the liquid sample. A wire wound on a drum attached to the axis, carries a mass, which accelerates the rotor when falling through a distance H in the time interval T. When completely unrolled the wire is rewound on the drum and the mass is lifted up to a smaller height H′ in a shorter time T′. The determination of both distances and time intervals permits calculation of the coefficient of viscosity.

A new Atwood's machine attachment has been designed in which one side of the system consists of a drum around which the string is wrapped so that the drum rotates as it falls. The other side of the system consists of a falling body, whose mass can be varied. Two sparktimer records are taken, one for the drum, and one for the falling body. Data taken are used to verify the law of motion of the mass center and the law of motion about the mass center.

A simple experiment is described for demonstrating the whip effect in a falling chain. The observed reduction in fall time due to the concentration of energy in the falling section of the chain agrees well with the assumption of small energy loss in the loop of the chain. The measured tension at the support point rises far above the maximum value which would be reached in the absence of energy concentration.

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