The Physics of War (5 page)

In many cases we are not interested in the work (or energy) that has been done, but rather the rate of the work, or the amount of work accomplished per unit of time. This is referred to as
power
. In mks units, power is measured as joules per second, and by definition a joule/sec is a watt.

ANGULAR MOMENTUM AND TORQUE

Another type of motion that is important in relation to warfare and weapons is rotational motion. The wheel, or anything that spins about an axis, has angular or rotational motion. And just as we have linear velocity and linear acceleration, we also have angular velocity and angular acceleration. Angular velocity is measured as the number of revolutions per unit of time. Another common unit is the number of radians (rad) per unit of time, where the radian is 360/2π ≈ 57.3 degrees (360 is the number of degrees in a circle and π is the circumference of a circle divided by its diameter, which is 3.1416). Angular speed can, of course vary, and when it does it becomes angular acceleration. Its units are revs/sec
²
.
⁵

In the same way we have a concept analogous to force. It's the force that causes the rotational motion, and it is called
torque
. It has to be applied at some distance from the rotational axis to cause rotation, so there is also a distance involved. Torque is defined as force × distance (f × r). Note that you apply torque every time you use a wrench or open a door.

Earlier, in the case of translational or linear motion, we also had momentum, and in the same way we have angular momentum in this case. To determine the formula for it we must replace mass (m) and velocity (v) by appropriate angular quantities. Velocity is no problem; we merely replace it with angular velocity (ω), but m is a little more complicated because we are dealing with a large number of small masses, each at a different distance from the axis. If we add up all the little contributions from these small masses we can determine what is called the moment of inertia; it is designated by I. Angular momentum is then Iω.

MACHINES

Many of the early weapons were what we call machines in physics. A machine is a device that makes work easier. A simple example of a machine is a long board used to raise a box that is too heavy for us to lift. If you place one end of the board under the box, and place a block (called a fulcrum) a few feet away, then apply a downward force to the other end of the board, you find you can easily raise the box. This makes sense because work is force × distance; when we apply the force the box is raised a smaller distance than the top of the board moves. We are, in effect, using the extra distance to get a greater force. The work done is equal, but it is easier for us because we only have to apply a fraction of the force we would have to if we were lifting the box directly. This is basic to all machines.
⁶

Many types of machines exist, and the principles associated with each of them were used in various early weapons. Some of the more common machines are:

Pulleys
: They allow heavy loads to be lifted with less force; you merely have to move the rope a greater distance than the load moves.

Wheel and axle
: A longer twist at the outer edge of the wheel exerts a more powerful but shorter motion near the axis.

Screw
: Applying a larger but easier rotary force creates a smaller forward motion.

PHYSICS OF THE BOW AND ARROW

The bow and arrow was used extensively in early warfare. Archers were trained from an early age. In some cases they advanced toward the enemy on foot, carrying a shield; in others cases they rode in chariots. As we saw in the
last chapter
, chariots usually had a driver and an archer, and when the chariot got close enough to the enemy, the archer would begin firing arrows as fast as he could.

A bow, in essence, is a simple machine that changes one type of energy into another, making it easier for the archer to give the arrow a high velocity. What is needed for a high velocity is a rapid and forceful arrow movement, and of course muscles can do both of these, but not at the same time. To understand the physics of the bow and arrow let's begin with the archer loading an arrow and pulling the string back slowly. He is using his arm muscles to do this. He pulls the string back to its maximum extension, and in the process the bow bends. The energy from the archer's muscle contraction is stored in the bending of the bow. This is potential energy. He then lets go of the bowstring, at which point the string moves rapidly to the normal rest position. In the process it transfers energy from the bow to the arrow. In essence, potential energy is transferred to kinetic energy, as in the case of a falling ball. The transfer is obviously very rapid, and this gives the arrow a high speed. Note that the archer has produced a certain amount of energy, and by the conservation of energy, this energy must remain constant. But the bow can move with both a high force and a high velocity in a way that his arm cannot. The bow is a machine that stores energy. Muscle power is used to load the machine at low speed, then the machine releases the energy at high speed. Indeed, if you know all the variables, such as the mass of the arrow, the distance the bow is pulled back, and what force it exerts, you can equate potential energy to kinetic energy and determine how fast the arrow will leave the bow. Furthermore, if you know the angle at which is it is aimed (and ignore air pressure) you can determine how far it will go.
⁷

Over the years bow and arrows were gradually improved. Several factors are important in relation to how powerful a bow will be. Three of the most important are its length, its shape, and its composition. In general, the longer the bow, the more powerful it will be, but other factors play a large role. We will see later that the English developed a very effective longbow and used it with considerable success against the French.

The overall shape of the bow is also important. Early bows had a single curve and were made of wood. Eventually, however, archers determined that if the ends of the bow were curved away from the user, the arrow would go farther. This was because the curving shortened the distance between the bow and the string at rest, and as a result, the string traveled farther before coming to a stop as it released the arrow. This extra push gave the arrow a little more momentum and speed. This type of bow is called a recurved bow.

The bow's composition was, of course, also critical. The type of wood, or other material, it was made of had a large effect on its power. Also, a bow's density, elasticity, and tensile strength (amount of stress it can take before it breaks) determines how much energy it can store and how well it returns to its original shape after the shot.

Early on it was discovered that bows made from more than one material were more effective than simple wood bows. They are referred to as composite bows. Composite bows were usually made of wood, a section from the horn of an animal, and sinew. A thin section of horn was glued to the belly of the bow on the side facing the archer. Horns from antelope, water buffalo, and sometimes sheep or goats were used. This allowed more energy to be stored in the bow. The glue was made from fish oil. Strips of sinew were also glued along the back of the bow, again to increase the energy storage. The tips (recurved sections) were also stiffened using sections of bone.
⁸

Arrows were continually improved over the years. One of the most critical concerns was the weight of the arrow. If an arrow was too light it would be affected by the movement of air and would not stay on course well. If it was too heavy, on the other hand, it would create a lot of drag and fall too fast. The ideal weight was somewhere in between. It was also discovered early on that feathers along the sides increased an arrow's stability, and that the length and height of the feathers had an effect on how far the arrow went, and on its stability in flight.

A variation on the ordinary bow and arrow is a crossbow, which is known to have been used by the early Greeks. It fired a steel bolt, and initially the drawstring had to be drawn back by the archer and locked into position, then released using a trigger. This made loading slow, and considerable strength was needed to pull the string back. Eventually, however, a mechanical winch system was developed for loading, and with it, a much greater tension could be put on the string. As a result, the steel bolt left the bow with a much greater velocity, and it therefore had a greater range. The crossbow could, in fact, project a bolt up to five hundred yards. But the problems didn't go away. The steel bolts were not very aerodynamic, and as a result they were also not very accurate. In addition, they were much slower and much more difficult to load than an ordinary bow. At maximum, a crossbow could be fired about twice every minute whereas a good archer could fire twelve to fifteen arrows a minute. Initially, however, crossbows had a serious advantage over the ordinary bow: the steel bolts they fired could penetrate the steel armor shielding of the enemy. Furthermore, the
bolts could easily kill a horse. Eventually, however, the English invented the longbow, which also packed enough power to penetrate armor.

A crossbow.

Although it was not used in early wars, a significant advance was eventually made in bows. A bow is, of course, hard to draw back, but the greater the energy you expend in drawing it back, the greater the energy it transfers to the arrow. And again machines came to the rescue. Pulleys were eventually used to help archers do more work on the bow (and produce more potential energy) with less physical effort. The compound bow allows an archer to hold and aim a drawn bow without a lot of stress or fatigue.

PHYSICS OF OTHER EARLY WEAPONS

Most of the early weapons were devices of one sort or another that projected either arrows or stones. Most were, in fact, catapults of some sort. As we've already discussed, three of the best known were the ballista, the trebuchet and the onager. The ballista was a torsion spring, which stored energy in several loops of twisted skeins. It could project heavy iron-tipped darts or stone projectiles of various sizes. The darts were placed in a shallow wooden trough or groove. The ballista was armed by hooking a bowstring behind the dart and winding it back using a windlass. It pulled a sliding trough and the dart within it back, and at the same time it twisted the skein (the bow string was attached to two arms, each with its own twisted skein). Ratchets and cogs prevented it from shooting while it was being loaded. Once loaded it could be fired using a trigger. Again, we have potential energy of torsion in the twisted skeins, and when it was released, it was converted to the kinetic energy of the dart.

The earliest ballista were developed about 400 BCE. The best ones had a range of about five hundred yards. They used only relatively light projectiles, so they didn't hit with a lot of force, but they were relatively accurate.

The trebuchet was much more powerful, and it worked on a completely different principle. It is, in fact, sometimes called the “counterweight trebuchet” because it used a counterweight to produce the energy for throwing the projectile. It is said to have first been used by the French in the twelfth century, and is based on the principle of the fulcrum described earlier. Its major part is a long arm anchored above the ground close to one end. At the end of the long arm is a sling that contains a pouch; this is where the projectile (usually a large stone) was placed. The energy came from a heavy weight placed at the end of the shorter arm. It was raised up and held in place until ready. Again, we have potential energy being converted into the kinetic energy of a projectile. When the trigger was released, the sling and longer arm would swing upward toward a vertical position. At this point the sling would release and the pouch would open. The projectile would be thrown forward with considerable velocity. The device was, in effect, powered by gravity.

The advantage of the trebuchet was that it could project stones of up to three hundred pounds, which could do considerable damage to the upper parts of most castle walls. It had a range of about three hundred yards. It's important to note that the sling played an important role; it could double the power of the trebuchet, allowing it to throw a projectile twice as far as it would have been able to without the sling.