LORNE A. WHITEHEAD OF VANCOUVER, B.C., offers a remarkable variant on a row of falling dominoes. Instead of having dominoes all the same size he has each domino 1.5 times larger than the preceding one. In a row of 13 dominoes a tap on the first domino, which is tiny, soon brings down the last, which is 64 times larger. If Whitehead had been able to continue scaling up the dominoes in the same way, the 32nd domino would have been as big as one of the twin towers of the World Trade Center in New York.

I began looking into how dominoes in a row fall using a row of 50 regular dominoes. At a separation of one domino width the chain reaction took two seconds. With about half that separation the reaction took half as long. I could hear the difference in the timing of the collisions in the two trials. The timing also changed noticeably after the chain reaction had passed the first five or six dominoes, presumably because in the early stages only a few dominoes were moving, whereas later more were falling at any one time.

By freezing the action with a snapshot or a strobe light I could see the pattern of the chain reaction. When the leading domino of the moving subgroup is falling toward the next domino, which is still stationary, the leading domino has the preceding domino leaning against it. Several other dominoes in the moving group are also leaning and falling. In a theoretical analysis of the chain reaction D. E. Shaw of Villanova University found that at any one time about five dominoes are in motion.

I next examined how the chain reaction ascends and descends a smooth ramp. The incline must of course be a shallow one or it will not be possible to stand the dominoes on end. When the chain reaction traveled up a properly inclined ramp, the dominoes had to be spaced farther apart than dominoes in a level chain. A chain reaction going down the ramp was much faster and succeeded with any spacing less than the height of a single domino. The chain reaction could-also be made to climb a small flight of shallow stairs, but the reaction was quite slow.

A chain of dominoes can also be made to split into two chains. At the junction the last domino of the original chain falls against the first dominoes in the new chains. A chain reaction can also be made to turn, indicating that it is not necessary for one domino to hit the next one squarely.

To understand the physics of domino toppling I considered a conventional domino of height h, width w and depth d. What factors are responsible for the stability of the upright domino? If I am to topple it, how hard must I strike one of its broad faces in order to rotate it about an edge? Why are deeper (thicker) dominoes harder to topple? Why cannot a line of cubes (such as a child's blocks) be toppled in a chain reaction?

Consider an upright domino. Gravity pulls downward on every small volume of it. A simpler way of looking at the situation is through the concept of the center of mass, which is at the geometric center of the domino. The total pull of gravity (the weight of the domino) is said to operate there. The upright domino is stable because its weight vector points toward the support region.

In principle I can put the domino in another stable arrangement by tilting it on one of its bottom edges until the center of mass is over that edge. Although the weight vector again points toward the support region, the balance is precarious because even a small force destroys it, toppling the domino. To knock over a domino I strike it so that it is tilted completely through its position of precarious stability. (I make the assumption that the friction from the table is large enough to keep the domino from slipping. Could a chain reaction take place on a frictionless table?)

As the domino tilts it embodies two forms of energy. The kinetic energy depends on the speed of tilting The potential energy has to do with the height of the center of mass above the table. When I strike the domino, I impart kinetic energy, but as the center of mass rises the energy is converted into potential energy. What is the minimum amount of energy I could impart to the domino so that it just barely passes through the position of precarious stability? Initially the center of mass is at a height h/2 above the table. When it is at its highest point during the rotation, it is a distance r from the table (the distance between the center of the domino and one of the domino's bottom edges).

If the domino has a mass m, it has a weight mg (g representing the acceleration of gravity). The potential energy of the domino at any instant during rotation is the product of the domino's weight and the height of the center of mass above the table. Hence to lift the center of mass from a height h/2 to a height r I must provide the domino with an energy of mg(r - h/2). If my blow delivers less energy, the domino cannot pass through the position of precarious stability and will fall back to its original orientation. If I deliver more energy, the domino falls over.

The key to the instability of an upright domino lies in its shallow depth. Consider a domino of fixed height and width. If the domino is narrower and the weight are low. Only a little energy is needed to lift the center of mass through the position of precarious stability. A light tap of the finger is enough. If the domino is wider, all the other factors (r, the weight and the energy needed to lift the center of mass) are larger. A tap of the finger does not provide enough energy and does not topple a thick domino.

The rotation of a domino is caused by a torque that is the product of a force and a lever arm. A blow against the domino creates such a torque. Figure 6 demonstrates how to determine the lever arm. Extend the vector representing the force. Draw a line from the rotation edge to intersect perpendicularly the extension of the vector. This line is the lever arm. In order to deliver a large torque that will make the domino rotate rapidly you should hit high on the face of the domino to have the benefit of a large lever arm. As the domino rotates, its weight vector creates another torque that attempts to return the domino to its initial orientation. To find the lever arm for this torque extend the vector representing the weight. Draw a line from the rotation edge to the extension of the vector, intersecting the vector extension per pendicularly. This line is the lever arm associated with the weight of the domino. As the domino rotates upward the lever arm is shortened, decreasing the torque imparted by the weight.

In the illustration the torque from the striking force acts to generate a clock wise rotation of the domino, whereas the torque from the weight acts in the opposite direction. Since the striking force is brief, its torque is also brief. During the ensuing rotation the only torque on the domino is from its weight. If you are to topple the domino, the torque you apply must provide enough rotation to prevent the weight's torque from stopping the movement before the center of mass passes through the position of precarious stability.

A tap of a finger on a child's block is unlikely to topple it. Since the block is deep, the initial lever arm for the weight's torque is large. Moreover, the block weighs considerably more than a domino of the same width and height. For these reasons the torque from the weight easily overpowers the torque from the striking force.

I then examined the matter of knocking over a domino so that it would collide with a second one. Obviously the distance between the dominoes must be less than their height or they will not collide. Is there a minimum spacing? If you strike the first domino vigorously enough, the second domino is bound to fall as long as the separation is less than the height of one domino. Suppose your blow barely pushes the first domino through its position of precarious stability. The second domino will fall only if it is beyond a certain distance (call it the least distance) from the first one. I did experiments with dominoes of two sizes. One set consisted of standard plastic dominoes 4.4 centimeters high, 2.2 centimeters wide and .7 centimeter deep. The other set consisted of large wood dominoes 13.9 centimeters high, seven centimeters wide and 1.9 centimeters deep. The width of a domino does not matter except as it contributes to the weight. The ratio of height to depth, which does matter, was about the same for the two sets.

I taped a sheet of fine-grain sandpaper to a table and taped a ruler to the sand paper. The sandpaper provided enough friction to keep a domino from slipping. Along the edge of the ruler I stood two of the smaller dominoes upright. Leaving the position of the second domino unchanged, I varied the distance between the two by moving the first one. In each trial I released the first domino from its position of precarious stability with essentially no kinetic energy. To put the domino into that orientation the separation between the dominoes had to be at least .7 centimeter. To make the first domino knock over the second the separation had to be at least 1.2 centimeters. At separations between .7 centimeter and 1.2 centimeters the first domino ended up simply leaning against the second, which remained upright.

The results with the larger dominoes were similar. To release the first of two dominoes in its position of precarious stability the separation had to be at least 1.9 centimeters. To topple the second domino the separation had to be at least 2.3 centimeters.

The least distance is determined by energy. To topple the second domino the collision must impart enough energy to lift the domino's center of mass through the position of precarious stability. This crucial amount of energy is mg(r - h/2). Assume an ideal collision in which at the instant of collision all the kinetic energy of the first domino is transferred to the second. How far must the first domino fall from its highest point in order to impart enough energy to the second domino to topple it? The first domino must fall just as far as the center of mass of the second domino must rise. Thus it must fall by a height of (r - h/2).

Assuming a complete transfer of energy in the collision, this requirement sets the value of the least distance between the dominoes. When the spacing is too small, the first domino cannot fall far enough to provide the necessary energy. In such cases the second domino rotates slightly but then rocks back to its initial orientation.

How does the value of the minimum distance determine the success of a chain reaction in a row of dominoes? Suppose the spacing in the row is so close that the first domino cannot be put into its position of precarious stability before it hits the next domino. The initial blow must be quite vigorous, since in essence it must make the entire chain lean as one.

If the dominoes are farther apart but still are separated by less than the least distance, a sound blow on the first domino is needed to give it enough kinetic energy after it has passed through its position of precarious stability. The additional energy is required because the domino will not be able to fall far before it collides with the next one. The second domino topples not because of the energy in the fall of the first one but because of the extra energy in the original striking force.

The least distance is important when the first domino has just enough energy to pass through its position of precarious stability. Then the second domino must be beyond the least distance so that the first one can fall far enough. Only with such a fall can enough energy be delivered to the second domino to topple it. A weak blow on the first domino can initiate a chain reaction when the dominoes are separated by more than the least distance and less than the height of a domino. If the separation is less than the least distance, the initial blow must deliver substantially more energy to the first domino to create a chain reaction.

With my home computer I estimated the values of the least distance for dominoes of various shapes. I assumed that the collision completely transferred the kinetic energy of one domino to the next. I also assumed that afterward the lean of the first domino helped to push the second one through the position of precarious stability. To find the least distance I had the first domino pass through its position of precarious stability with no kinetic energy. To impart enough energy to the second domino the center of mass of the first has to fall to a height of h/2.

My estimate of the least distance for my small dominoes was .7 centimeter and for the larger ones 1.9 centimeters These estimates were below my experimental results of 1.2 and 2.3 mainly because my calculations assumed a perfect transfer of kinetic energy in the collisions. In a real chain reaction some of the energy is lost to vibration of the dominoes and to friction as they touch each other. When I lubricated the faces of the dominoes, the experimental results decreased only slightly, implying that vibration contributes more than friction to the loss of energy.

I glued some of the smaller dominoes together to make new dominoes two or three times deeper than the originals. Repeating my experiments, I found that the doubly deep dominoes required a least distance of separation of 2.2 centimeters. (My computer estimate was 1.5 centimeters; energy losses in the real collisions again accounted for the difference.) The triply deep dominoes almost toppled one against the next, but friction held them at the last instant, leaving them at a tilt.

My calculations showed that the necessary least distance in spacing increases as the depth of the dominoes increases. The reason is that each domino must fall farther in order to gain enough kinetic energy to topple the next one. When the required least distance is almost equal to the height of a domino, the energy transfer is inefficient and the chain reaction stops. A line of cube blocks cannot topple in a chain reaction because the required least distance exceeds the height of the blocks.

Next I returned to some of my earlier demonstrations. Normally a chain reaction of dominoes is initiated with a vigorous rap on the first domino. Each domino in the chain passes through its position of precarious stability with more than enough energy. Hence even when the dominoes are spaced closely and have little chance to fall, the collisions move them safely through their position of precarious stability. The kinetic energy in the wave is large enough for the wave to split at a junction and then travel along two chains.

A chain reaction can be set off on a ramp provided the ramp is not too steep. The weight vector must point toward the lower face of the-domino. You will also find that the dominoes must be separated more than they need to be on a level surface.

Consider the second domino in the chain. Since the ramp slants, the center of mass must be raised quite far in order to pass through the position of precarious stability. Thus the domino is slower to topple than one on a level surface. Toppling requires more energy. A domino must fall a substantial distance for there to be a significant transfer of energy when it strikes the next domino. Therefore the least separation between dominoes is large and the chain reaction moves slowly.

A chain reaction moves faster down a ramp. Each domino needs only a small amount of energy in order to pass through its position of precarious stability. The least separation is essentially zero. Moreover, each domino does not have to fall far to topple the next one, so that the time from-one toppling to the next is quite brief.

The dominoes on a staircase topple slowly. Since each domino strikes the next one low on a broad face, the lever arm for the collision is small, providing a small torque. The transfer of energy is also inefficient. The struck domino rises slowly through its position of precarious stability, keeping the speed of the reaction low.

Whitehead's demonstration of consecutively scaled-up dominoes is stunning. The first domino is so small that can hardly stand it upright. The last on (the 13th) is so heavy that I have difficulty lifting it into place. Yet the chain reaction initiated with the tiny domino easily drops the heavy one.

Since each domino is 1.5 times large in every dimension than the preceding one, the last is not only 64 times larger than the first but also 262,144 times heavier. With a center of mass 64 times higher than that of the first domino its potential energy is almost 17 million times greater. A slight nudge on the first domino gives it about .024 microjoule of kinetic energy. The kinetic energy of, the last domino as it completes its fall is !$ about 51 joules, which is two billion w times the amount imparted to the first domino.

Whitehead made his dominoes out of acrylic sheets, sandblasting them to make them smoother. He laminated several thin sheets to make the larger dominoes. When he lines up the group of 13 dominoes for a chain reaction, each domino is separated from the nextlarger one by about its own width.

Could a chain of dominoes be designed with a scale factor larger than 1.5? Assuming an ideal transfer of energy in the collisions, I think a scale factor of about 2.5 is the limit for dominoes shaped like Whitehead's. Larger scaling iD factors are possible if the ratio of height to depth is increased. You might like to experiment with this problem.

Here is a puzzle. The object is to stack dominoes at the edge of a table so that the stack leans out from the edge as far as it can without falling to the floor. Each domino is positioned with a broad face down and its long dimension at right angles to the edge of the table. How should the dominoes be laid so that the top one is displaced from the table's edge by the maximum amount? What is the smallest number of dominoes needed to make the top domino completely clear the edge of the table? Is there a limit to how far the stack can extend from the table?

A problem of this kind was posed in 1955 by Paul B. Johnson. His solution was later simplified by Leonard Eisner, who also applied it in a prank. One night Eisner and another graduate student constructed a leaning stack of a library's volumes of The Physical Review, leaving the disconcerting overhang as a surprise for the librarian. (They were presumably too young to have learned that if you need help from librarians, or even if you want to avoid being fired from the university, you had better not fool around with the library volumes.)

Consider the first domino put down. It is stable as long as its center of mass is over the table so that the weight vector cannot supply a torque. Its maximum stable overhang is achieved when its center of mass lies almost over the edge.

Now you put a second domino under the first. The outer edge of the lower domino acts in place of the edge of the table to keep the top domino from rotating. How should this pair be placed on the table? The combined center of mass of the pair, which lies midway between the centers of the dominoes, must be placed over the table's edge to achieve the maximum stable overhang. No computation is needed. A few trials will reveal the proper balance.

When a third domino is put under the first two, the combined center of mass of the three dominoes must be above the edge of the table. This procedure can continue.indefinitely. With a stack of n dominoes the distance between the edge of the table and the outer edge of the top domino is given by the series (h/2) (1 + 1/2 + 1/3 + 1/4...1/n), where h is the long dimension of a domino and the expression in the second set of parentheses is the harmonic series.

I programmed my computer to sum the expression to any wanted number of dominoes. At least four dominoes are required if the overhang is to exceed h, making the top domino fully clear the table. To have the top domino clear the table by two domino lengths you must stack 31 dominoes. Thereafter the number goes up rapidly. You will need 12,367 dominoes to make the top domino clear the table by five domino lengths and 1.5 X 10⁴⁴ to make it clear by 50 domino lengths.

Evaluating the expression for the maximum overhang is simple enough for a computer until about a million dominoes are being considered. A computer's normal accuracy (called single precision) is usually limited to seven significant figures, six figures are printed out after the computer has rounded off the calculation. To continue the overhang expression past one million dominoes leads to error unless the program calls for double precision (usually an accuracy of 14 significant figures, with 13 figures printed out after rounding off). Computation with a large number of dominoes at double precision is slow. Even with this precision the computer's arithmetic functions may yield only approximate results.

Steve Wallin of Laramie, Wyo., has shown me a fast technique for approximating the overhang expression. It employs as a benchmark the 1,674 dominoes needed for an overhang of four domino lengths. The number of dominoes needed for a given overhang (expressed in domino lengths) is equal to 1,674 multiplied by the exponential of twice the difference between the wanted overhang and four lengths. For example, if your objective is an overhang of 10 domino lengths, subtract 4 from 10, multiply by 2 and take an exponential of the result. Finally multiply by 1,674. The answer is the number of dominoes you need.

Here is another puzzle. What is the maximum amount by which a domino can be made to clear the edge of the table when you are working with only three dominoes? Two solutions are shown in the illustration above. In the first solution the center of mass of the lower domino is just above the edge of the table. The second domino is put on the inner end of the first; the upper domino's center of mass is just over the edge of the lower domino. The third domino is now put on the outer end of the first domino. The combined center of mass lies just above the edge of the table, so that no torque is applied to the pile by the weight.

A better solution is to rotate the first domino so that a diagonal line running between two of its corners is at right angles to the edge of the table. The second domino is balanced at a corner distant from the table's edge. The third domino is balanced with its center of mass over the opposite corner. Again the combined center of mass lies at the edge of the table, giving the assembly stability. With this arrangement the third domino is farther from the edge of the table than the third domino in the other arrangement because the diagonal line across the domino is longer than a domino length. You might try standing the third domino on one of its narrow side faces. The center of mass is in the same relation to the table as before but the nearest face of the domino is farther from the edge of the table.

Bibliography

LEANING TOWER OF LIRE. Paul B. Johnson in American Journal of Physics, Vol. 23, No. 4, page 240; April, 1955.

DOMINO "CHAIN REACTION." Lorne A. Whitehead in American Journal of Physics, Vol. 51, No. 2, page 182; February, 1983.

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