THE CONNECTION between a falling chimney and a breaking pencil point is far from obvious, but in most of what follows I shall undertake to make the connection clear. Then I shall take up a quite unrelated matter: an ingenious device that couples a telescope and a pocket calculator in such a way that changes in the angle at which the telescope is pointed are recorded by the calculator and can be directly read off the calculator's display panel. If you ever have an opportunity to watch as an old chimney (the tall, free standing and basically cylindrical kind) is demolished, look carefully at one side of it as it begins to fall. In the usual demolition procedure a section of the base is knocked out by dynamite or a bulldozer, forcing the chimney to fall to one side. By the time it has reached a tilt of about 45 degrees it will probably show a lateral crack near its middle. As a result the top part of the chimney begins to fall more slowly than the bottom part, and the two parts form a broad V. The reason for the break is that as the chimney begins to fall the top part necessarily has to accelerate downward more rapidly than the bottom part. The chimney is usually not strong enough to withstand the bending stresses and the acceleration, and so it cracks.

Ernest L. Madsen of the University of Wisconsin Medical School has recently examined the mechanics of a falling chimney. He first considers the forces on the entire chimney once it is falling. It is subjected to three forces: its own weight (acting through the center of mass), an upward reaction force from the ground and friction (horizontally) from the ground. The combination of these forces makes the chimney accelerate toward the ground and rotate vertically at the base.

Madsen next considers a lower section of the chimney extending from the base to some arbitrary point along its length. This section has an angular acceleration around the base throughout the fall because of three torques acting on it. One torque arises from the sections own weight. The other two torques arise because the lower section must drag the upper section around in the rotation. One of these torques is due to the shearing force resulting when the upper section attempts to slide over the top of the lower section. Finally, the lower section is subjected to a bending torque because the upper section tends to lag behind in the rotation and so to bend backward.

Except for the exact bottom and top of the chimney any theoretical choice for the length of the lower section will include a bending torque acting to sever the lower section from the upper one. According to calculations, the bending torque is greatest at a point a third of the way up the chimney. As the chimney begins to fall and to be subjected to an angular acceleration around the base, the bending torques along its entire length increase with time. Eventually the bending torque at the point a third of the way up the chimney is enough to rupture it, so that the upper two-thirds breaks off and begins to lag behind the lower third.

Madsen's results pertain to a chimney that is uniformly cylindrical. In a chimney that is tapered or has some other form the maximum bending torque will be at some other point along its length. If you are able to watch a chimney being toppled, you might try to photograph the break. It would be particularly interesting to photograph the action in slow motion so that you can see the crack begin in the lower section and then propagate across the width of the chimney. (Stay alert. When I watched a falling chimney recently, I had to dodge bricks that were thrown into the air when the chimney hit the ground.)

The rupture does not travel directly across the width of the chimney. In 1940 Francis P. Bundy explained the rupture pattern as being due not only to the bending stress but also to the accompanying compression of the chimney on the trailing side. The bending apparently forces the line of maximum compression downward as the rupture travels to that side.

Some of Bundy's calculations suggest that if a tall chimney does not break during its fall, something strange may happen: its base may hop up into the air near the end of the fall, apparently because the chimney is rotating about its center of mass. This result is not predicted by Madsen's calculations. You might want to watch the base of a chimney that falls without breaking to see if it does in fact hop.

A falling chimney can display two other interesting features. One is another break point that may develop close to the base because of shearing as the top part of a tall, heavy chimney attempts to slide over the lower part. Some photographs of falling chimneys clearly reveal the extra breaking: the chimney breaks in two places.

The other interesting feature is that the base of the chimney may slide during the fall. I would have guessed that such a slide would be in the direction opposite to the direction of fall, but normally it is not. The base moves the way the chimney does, being dragged along in the direction in which the chimney has acquired some horizontal momentum. According to Madsen's work, the base is most likely to slide in this way when the chimney has fallen through at least 50 degrees. Beyond that angle the amount of friction needed to hold the base in place becomes impractically large.

The falling chimney can be modeled somewhat imperfectly with an unsharpened pencil that is placed vertically on a level surface, eraser end up, and then allowed to fall. The lower end does come to rest displaced in the direction of the fall, but the motion is more than just a simple slide. The end first moves a short distance in a direction opposite to the direction of fall, but as the pencil bounces on the surface, the end hops a greater distance in the direction of fall. The static friction on the lower end of the pencil is apparently not enough to keep that end from sliding along the surface as it tends to rotate about the falling center of mass. The later motion in the direction of fall results from the lower end's being dragged in the direction in which the falling pencil has acquired horizontal momentum.

Madsen has suggested to me a simple arrangement of Tinker Toy components with which you can simulate the bending of a falling chimney. A Tinker Toy set includes a squat wood cylinder with a central hole through it. You need a stack of about 30 of them to simulate the chimney. A Tinker Toy stick fits snugly through the bottom cylinder. Sharpen the lower end of the stick so that the stack will be held in place when you push the point into a small piece of sponge. Run an elastic band, preferably a latex one, around the sharpened stick and up through the central holes in the cylinders. You will probably need several of these bands looped together to reach the top of the stack. Place across the hole of the top cylinder a smaller stick around which the top band is looped. The cylinders should be aligned to make a straight "chimney," and the bands should not be tangled inside the cylinders. The tension should be adjusted so that the chimney bends if it is leaned to one side.

The pointed end of the lower stick is stuck into the sponge, which is glued or clamped to a large board held stationary on a tabletop. When the chimney is pushed gently to one side and falls, it bends backward much as a real chimney does. The base does not slip away from the direction of fall because of the friction that arises between the stick and the sponge.

Since the fall is rather quick, a photograph of the falling stack is more revealing than direct observation. Madsen suggests that motion pictures made at 36 frames per second or faster are best. If you lack the equipment for that, you can make a snapshot of the chimney with a strobe light. Turn off the room lights, open the aperture of the camera, cause the chimney to start falling, flash the strobe light during the fall and then close the aperture before turning on the lights again.

A common demonstration apparatus in physics classes is related to the falling chimney and illustrates the large acceleration at the top of a chimney that does not crack. A board is hinged at one end to a base that sits on a tabletop. The other end of the board is propped up with a stick so that the board makes an angle of about 35 degrees with the tabletop. A small paper cup is fastened to the upper side of the board near the free end. Still closer to the free end is a holder for a small metal ball. The holder could be just a notch in the board. The positioning of these items is such that when the stick is knocked away, the ball separates from the board (because the board accelerates downward faster) and falls into the cup.

One might expect the ball and the board to fall at the same rate, in which case they would reach the tabletop at the same time and the ball would never end up in the paper cup. Once the stick is knocked away the ball is in free fall; it accelerates downward with the normal gravitational acceleration of about 9.8 meters per second per second. The board is subjected to an angular acceleration around the hinge because the weight of the board creates a torque that acts on the center of mass. This motion causes the upper end of the board to accelerate downward at a rate exceeding the acceleration caused by gravity. The rate does not depend on the length of the board (because the center of mass will still lie at the midpoint of any length and the torque there will give rise to the same acceleration at the upper end), but it does depend on the angle between the board and the table. As the board falls and the angle decreases, the downward acceleration of the upper end increases to a value of 1.5 times the normal acceleration. As a result the board and the cup reach the tabletop before the ball does. If everything is positioned correctly at the outset, the ball will drop neatly into the cup.

Albert A. Bartlett of the University of Colorado at Boulder has recently described a modification of this design. He fastened a weight at the uppermost end of the board, just above the notch for the ball. I would have thought that the additional weight would cause the board to accelerate downward even faster, but actually it has just the opposite effect. The additional mass on the board slows the acceleration because the torque cannot accelerate both the board and the additional mass as much as it does the board alone. If enough extra weight is added, the board cannot reach the table ahead of the ball and the ball therefore does not drop into the cup. Under some circumstances the acceleration of the board is so slow that the ball remains in contact with the board at the beginning of the fall and acquires a horizontal velocity before it finally separates from it. The ball then overshoots the cup instead of landing in it.

Now for the pencil. As you are quite aware, the point of a pencil is likely to break if you push down too hard while you are writing. The point cracks because of bending stresses, in much the same way a falling chimney does.

Have you ever noticed that all broken pencil points are about equal in length? Donald H. Cronquist of San Jose has observed this consistency and has devised a simple model to explain it. He first ruled out defects of manufacturing and damage from sharpening as the primary cause of the uniform breaks. Manufacturing defects would be too random from pencil to pencil, and damage from sharpening could be detected by looking closely at the point. It seemed more likely that the point ruptured on its bottom side at a place where the bending stress exceeded the tensile strength of the lead Tensile strength is a measure of the maximum stress (the force per unit of area) a material can withstand without bending as the stress acts to elongate it When the stress exceeds that maximum value, the material ruptures. (If, on the other hand, the material is compressed, the stress ultimately exceeds the compressive strength of the material and the material collapses.)

When you write with a pencil, the forces on the tip from the paper and the tabletop set up stresses along the point. In particular a net force perpendicular to the axis of the pencil acts to bend the point, compressing the top side and elongating the bottom side. Stresses arise across a cross section of the point. In a brittle material such as pencil lead the maximum possible stress in elongation (the tensile strength) is usually less than the maximum possible stress in compression (the compressive strength). Hence when a stress limit is reached, it is most likely to be at a place on the bottom side of the point.

When a pencil is sharpened, the tip is normally not made perfectly sharp and so the point does not form a complete cone. A certain length is missing. Cronquist generalized his calculations by measuring the distance from the actual writing tip upward as a ratio of the missing length. His calculations indicate that the bending stress is at a maximum at a place distant from the writing tip by half the missing length. Since the sharpener grinds the point in a conical shape, this result means that the bending stress is at a maximum at a place where the cross-sectional diameter is 1.5 times that of the writing tip (a ratio of 3 to 2). An idealized pencil is most likely to rupture at that place.

For a blunter tip the broken-off point is longer, because with a larger diameter for the writing tip the place of maximum bending stress is farther up the cone formed by the point. Since the point characteristically breaks soon after the pencil is sharpened, however, the broken-off points will usually be short and of about equal length. The actual force and stress required for breaking will vary from pencil to pencil, depending on the hardness of the lead and the angle of the cone on the pencil point, but the maximum bending stress should still occur at the theoretical place determined by the 3 :2 ratio of diameters.

The analysis does not extend to severely blunted pencils. In such a pencil the breaking point would theoretically be up on the wood shaft. Moreover, the 3 :2 ratio of diameters representing the most likely place of breaking may not exist. The analysis also does not allow for variations that actually are due to manufacturing and sharpening and does not consider the variety of forces on the pencil during use. Finally, the analysis ignores the net force parallel to the axis of the pencil, concentrating instead on the net force perpendicular to the axis. Nevertheless, the formula does predict the lengths of the broken-off points fairly well.

I checked the analysis by systematically sharpening and breaking the points of several No. 2 pencils. The diameters and lengths of the broken-off points (measured from the writing tip to the place where the rupture began on the bottom side) were measured with a micrometer. At first I made no attempt to standardize the breaking. The results were inconclusive because the lengths of the broken-off points varied considerably even when the tip diameters were the same.

Next I tried to make the breaking more consistent. When I sharpened a pencil, I rotated it as I rotated the handle of the sharpener in order to make the grinding uniform around the pencil point. Then I held the pencil so that its tip was at a certain place on the table and my hand was poised just above a rubber stopper I used as a reference height. In this way I kept the angle of the pencil with the tabletop roughly the same (about 45 degrees) throughout. With a pencil so positioned I pressed down until the point snapped off. (Take care that the point does not fly into your eye.) If the rupture occurred up on the wood casing around the lead, I discarded the datum.

The points nearly always broke in the same way. The rupture began on the bottom side of the point and propagated upward and away from the writing tip until it reached the top side. Neither Cronquist nor I know exactly why the rupture propagates in this manner, although the pattern is similar to the rupture patterns of falling chimneys.

I plotted the lengths of the broken-off points against the diameters of the writing tips. To fit a straight line through the data points I used a pocket calculator that could do a linear regression. With both axes on the graph having the same scale, the slope of the line was 2.5. With my sharpener, which produces pencil points whose apex angle is about 12 degrees, the slope of the line should be about 2.2 according to Cronquist's theory. Considering the variation in the construction of pencils the experimental results were surprisingly close to the theoretical ones.

You might continue collecting data on the lengths of broken-off points by investigating pencils of differing hardness and diameter. Is Cronquist's model correct in predicting that neither factor plays a major role in the lengths of the broken-off points? You might also see how the length varies as the angle between the pencil and the table is varied. For any such variation you need to collect a lot of data, plot them and then determine the best fit of a straight line through them. You might find that the theory does not work as well when the pencil tip is quite sharp or quite dull and that something other than a straight line fits the data better.

MECHANISMS for indicating the position of a telescope are sometimes awkward and difficult to read in the dark. John and Dave Guerra of Ludlow, Mass., have designed a position indicator employing the integrated circuit of a pocket calculator. The device is inexpensive, highly accurate and easy to read in the dark.

Not all calculators are suitable. You can find an appropriate one by the following procedure. Turn on the calculator and press first the "plus" key and then the "1." Next press the "equals" key several times. The calculator should add 1 to a running sum each time you push the equals key. Now press the "plus-minus" key once (this key is the one that causes the calculator to change sign) and the equals key again several times. The calculator should subtract 1 from the sum with every push of the equals key. If the calculator does all of this properly, it is suitable for the Guerra design.

The Guerras' idea was to couple the rotating shaft of the telescope to the digital display of the calculator so that the display gives the orientation of the telescope. The coupling is accomplished in several steps. The Guerras' telescope has a polar driveshaft with a gear ratio of 10: 1. They attached a clock reduction mechanism with a ratio of 120: 1, thereby providing a net output rotation 1,200 times the rotation of the telescope. On the final drum of the clockwork they drilled six holes arranged to rotate through a light beam directed on a photocell. Each time the final drum of the clockwork rotated one revolution the 6eam fell on the photocell six times. One complete polar rotation of the telescope therefore resulted in 7,200 light pulses (6 X 1,200). The photocell was wired to the equals key on the calculator so that each light pulse actuated that button and added another I (or any other chosen quantity) to the running sum. The Guerras built two of these devices, one for the right-ascension axis of the telescope and one for the declination axis.

To use the devices the Guerras first orient the telescope toward a star of known position, setting their calculator at the star's coordinates as obtained from a star atlas. Then they turn the telescope about its rotational axes to another star whose position they want to determine. The rotation triggers the photocells, which in turn actuates the calculators. The new coordinates are then read from the displays.

The interval in a calculator (how much the calculator advances with each light pulse on the photocell) can be preset. Suppose you make it .2. Then each pulse increases the display by .2. A full rotation of the telescope, which corresponds to 24 hours, or 1,440 minutes, reads out as 1,440. In other words, the readout is in minutes of time, which would be suitable for right ascension. If one prefers minutes of arc (suitable for declination), a preset quantity of 3 would be appropriate.

The Guerras used a Concept V calculator in their setup. Other brands may suit you better. If you use another calculator, you will have to repeat the Guerras procedure for determining how the pins of the circuit are coupled to the keys on the keyboard. For the Concept V the Guerras were able to determine by visual inspection how the switches behind the individual keys connected to the leads on the integrated-circuit board and how the leads in turn were connected to the pins on the back of the board.

Visual inspection may be too difficult with some calculators. Then you will have to probe the pins with a device that consists of an inexpensive n-p-n transistor, some resistors and a push-button switch. Choose the resistor associated with the switch so that about 10 milliamperes goes through the resistor when the switch is closed. For example, if 10 volts is put through the switch (as in the bottom illustration on page 163), the resistor should be 1,000 ohms. When the switch is pressed, a signal is sent to the transistor, causing a positive logic signal to be fed from the battery of the calculator to the pin under investigation. If you have a calculator that responds to a negative logic signal, interchange the positions of the push-button switch and its resistor. Some calculators, such as the SR-40 made by Texas Instruments Incorporated, work differently. Instead of having a common ground on one pin, the calculator is designed so that when a key is pressed, it short-circuits two pins to actuate the desired process. With any calculator you should do your probing carefully in order to avoid damaging the instrument.

When the telescope is rotated in the direction opposite to what is considered the positive sense, the calculator has to be told to subtract each pulse it receives from the photocell. The Guerras devised a simple mechanism that triggered the plus-minus key so that the proper operation would be performed. A thin metal band (acting as a friction clutch) around the rotating drum holds a small magnet. Positioned around the magnet are three reed switches. When the telescope is rotated in the positive sense, the magnet rests against one of the reed switches, with the result that each light pulse causes the calculator to add. When the telescope changes direction, friction causes the metal band to rotate with the drum, so that the magnet passes the central reed switch and comes to rest on the third switch. When the magnet passes the central switch, the magnetic field momentarily closes the switch by forcing together the two leads in the switch. During that brief closing the switch sends a pulse to the calculator to switch signs. When the magnet comes to rest again, this time on the third switch, light pulses actuate the photocell and the equals key again, but now the calculator subtracts the preset interval from the preceding sum. The two outer reeds are in electric parallel with each other and together are in electric series with the photocell to prevent erroneous pulse counts during the change of direction.

The Guerras also constructed a turn-on mechanism that takes care of the initial operations of actuating the plus key and presetting the desired interval. It consists of a sliding, spring-loaded bar of Plexiglas fitted with a small magnet. Below the path of the magnet is another sheet of Plexiglas containing several reed switches. Each switch is wired to the calculator keys that would normally have to be pressed to prepare the calculator for use. To prepare the calculator one slides the bar over the reed switches. The last switch, which acts as a stop for the magnet, actuates the penlight cells and a three-volt flashlight bulb that shines on the photocell. If the operator wants instead to use the calculator just as a calculator, the magnet is pulled back short of this last reed switch.

Bibliography

STRESSES IN FREELY FALLING CHIMNEYS AND COLUMNS. Francis P. Bundy in Journal of Applied Physics, Vol. 11. No. 2, pages 112-123; February, 1940.

THEORY OF THE CHIMNEY BREAKING WHILE FALLING. Ernest L. Madsen in American Journal of Physics, Vol. 45, No.2, pages 182-184; February 1977.

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