The ancient Greek scientist Archimedes has a famous saying that has been passed down through the ages: "If you give me a fulcrum, I can move the earth!" This sentence has a strict scientific basis.
Archimedes first proposed the lever principle in his book "On the Balance of Plane Figures". He first regarded some empirical knowledge in the practical application of levers as "self-evident axioms", and then based on these axioms, using geometry through rigorous logical argumentation, he derived the lever principle. These axioms are:
(1) If equal weights are hung on both ends of a weightless rod at equal distances from the fulcrum, they will balance;
(2) On a weightless rod, If two ends of a weighted rod are hung with unequal weights at equal distances from the fulcrum, the heavy end will tilt downward;
(3) When the two ends of a weightless rod are at unequal distances from the fulcrum If equal weights are hung, the far end will tilt downward;
(4) The function of one weight can be replaced by the functions of several evenly distributed weights, as long as the position of the center of gravity remains unchanged. On the contrary, several evenly distributed weights can be replaced by one weight suspended at their center of gravity;
(5) The centers of gravity of similar figures are distributed in a similar way...
It is from these axioms and on the basis of the "center of gravity" theory that Archimedes discovered the lever principle, that is, "when two heavy objects are balanced, their distance from the fulcrum is inversely proportional to their weight. Archimedes The study of levers not only stayed in theory, but also made a series of inventions based on this principle. It is said that he once used levers and pulleys to successfully launch a mast parked on the beach to defend Syracuse. During the battle of the Roman naval attack, Archimedes used the principle of leverage to create catapults at long and short distances, and used them to shoot various missiles and boulders to attack the enemy. He once blocked the Romans outside the ancient city of Syracuse for three years. .
Introduction to the principle
The lever principle is also called the "lever balance condition". To balance the lever, the two forces (power point, fulcrum and resistance point) acting on the lever must be balanced. The size is inversely proportional to their moment arms. Power × power arm = resistance × resistance arm, expressed in algebra as F1? L1=F2?L2. In the formula, F represents power, L1 represents power arm, F2 represents resistance, and L2 represents resistance. Arm. It can be seen from the above formula that in order for the lever to reach equilibrium, the power arm is several times the resistance arm, and the power is a fraction of the resistance.
Conceptual analysis
When using a lever, in order to save effort, you should use a lever with a power arm that is longer than the resistance arm; if you want to save distance, you should use a lever with a power arm that is shorter than the resistance arm. Therefore, using a lever can save effort and distance. If you want to save effort, you must move more distance; if you want to move less distance, you must expend more effort. It is impossible to achieve this in the "center of gravity" theory based on these axioms. On the basis of this, Archimedes discovered the lever principle, that is, “when two heavy objects are balanced, their distance from the fulcrum is inversely proportional to their weight.
The fulcrum of a lever does not have to be in the middle. A system that satisfies the following three points is basically a lever: fulcrum, force application point, and force receiving point.
The formula is written as follows: distance from the fulcrum to the force point (moment) * force = distance from the fulcrum to the force point (moment arm) * force, this is a lever.
Leverage also includes labor-saving levers and labor-saving levers, both of which have different functions. For example, there is a foot-operated air pump, or a hand-pressed juicer, which is a labor-saving lever (force arm gt; torque); but we have to press a larger distance, and the force-receiving end only has a small movement. There is also a laborious lever. For example, on a roadside crane, the hook for fishing is at the tip of the entire rod. The tail end is the fulcrum, and the middle is the hydraulic press (torque gt; moment arm). This is a laborious lever, but in exchange for the labor, the middle force application point only needs to move. For a small distance, the tip hook will move a considerable distance.
Both levers have their uses, but where to use them must be evaluated to save effort or range of motion.
There is another thing called an axle, which can also be regarded as a lever application, but the performance may sometimes include rotation calculations.
The ancient Greek scientist Archimedes has a famous saying that has been passed down through the ages: "If you give me a fulcrum, I can move the earth!" This sentence is not only an inspiring aphorism, but also an inspirational aphorism. It has a strict scientific basis.
Lever classification
Leveres can be divided into labor-saving levers, labor-saving levers and equal-arm levers. These types of levers have the following characteristics:
1. Labor-saving levers: L1gt; L2, F1lt; F2, which saves effort and distance. Such as claw hammers, guillotines, etc. for pulling out nails.
2. Effortless lever: L1＜L2, F1＞F2, effortless and distance-saving, such as fishing rods, tweezers, etc.
3. Equal-arm lever: L1=L2, F1=F2, which is neither labor-saving nor labor-intensive, and does not move much distance, such as balances, fixed pulleys, etc.
Leveres in the human body
Leverages are indispensable in almost every machine. Even in the human body, there are many levers at work. Picking up an object, bending down, or even tilting your toes are all levers of the human body at work. Understanding the levers of the human body can not only increase your physical knowledge, but also learn a lot of physiological knowledge.
Nodding or raising your head relies on the action of a lever (see picture). The fulcrum of the lever is at the top of the spine. There are muscles in front and behind the fulcrum. The weight of the head is the resistance. The muscles before and after the fulcrum cooperate, some contract and some lengthen, and they cooperate to lower the head and lower the head. It can be seen from the picture that lowering the head requires less effort than raising the head.
When you bend your elbow to lift a heavy object, your arm is also a lever (as shown in the picture). The elbow joint is the fulcrum, and there are muscles on the left and right sides of the fulcrum. This is a kind of laborious lever. To lift a weight, the muscles will expend more than 6 times of effort. Although it is laborious, it can gain speed.
When you raise your toes, the muscles behind your heels are working. Your toes are the fulcrum, and your weight falls between them. This is a labor-saving lever (pictured), and the pulling force of the muscles is smaller than the body weight. And the longer the legs, the easier it is to use.
If you bend down, your muscles will exert a pulling force of nearly 1,200 Newtons. This is due to the fact that the lever formed between the lumbar muscles and the spine is also a laborious lever (pictured). Therefore, when bending down to lift a standing object, the correct posture is to keep the heavy object as close to the body as possible. To avoid muscle strain.
Discovery process
Archimedes first proposed the lever principle in his book "On the Balance of Plane Figures". He first regarded some empirical knowledge in the practical application of levers as "self-evident axioms", and then based on these axioms, he used geometry to derive the lever principle through rigorous logical argumentation. These axioms are: (1) Hang equal weights on both ends of a weightless rod at equal distances from the fulcrum, and they will balance; (2) Hang both ends of a weightless rod at equal distances from the fulcrum. If there are unequal weights, the heavier end will tilt downward; (3) If equal weights are hung on both ends of a weightless rod at unequal distances from the fulcrum, the far end will tilt downward; (4) The effect of a heavy object This can be replaced by the action of several evenly distributed weights, as long as the position of the center of gravity remains unchanged. On the contrary, several uniformly distributed weights can be replaced by one weight suspended at their center of gravity; the center of gravity of the figure is distributed in a similar way... It is from these axioms that on the basis of the "center of gravity" theory , Archimedes also discovered the lever principle, that is, "When two heavy objects are balanced, their distance from the fulcrum is inversely proportional to their weight."
Archimedes' research on levers is not just theoretical. aspect, and a series of inventions have been made based on this principle. It is said that he once used levers and pulleys to successfully launch a mast ship parked on the beach. In the battle to defend Syracuse from the Roman navy, Archimedes used the lever principle to create long and short slings, and used it to shoot various missiles and boulders to attack the enemy. He once blocked the Romans in Syracuse. It lasted for 3 years outside the ancient city.
It should be mentioned here by the way that there are records of leverage early in the history of our country. The Mohists of the Warring States Period once summarized the laws in this area, and there are two articles specifically recording the principle of leverage in the "Mo Jing". These two articles speak comprehensively about the balance of leverage. There are those with equal arms and those with unequal arms; there are those that change the weight of the two ends to make it deviate, and there are also those that change the length of the two arms to make it deviate. Such records are also very valuable in the history of world physics, and Mozi's discovery was about two hundred years earlier than Archimedes.
Historical Story
Archimedes locked himself in a hut and was working day and night on "The Theory of Floating Bodies". That day, a man suddenly broke in. As soon as he entered the door, he shouted: "Oh! Your old gentleman is hiding here." The king is mobilizing a large number of troops to look for you all over the city. ’ Archimedes recognized him as a court minister and thought, something big must have happened outside. He immediately packed up the parchment manuscript, reached out and grabbed a small round shell hat, went out with the minister, and went straight to the palace.
When they came to the front steps of the palace, they saw various carriages stopped. Guards with silver guns and iron helmets stood in two rows. The palace was full of civil and military people, and there was silence. The king was pacing back and forth on the carpet anxiously. Because the temple was dark, tall candlesticks were lit before it got dark. On the long table under the lamp are coastal defense maps and land defense maps. Looking at all this, Archimedes knew that the war he was most worried about had finally broken out.
It turns out that after the decline of ancient Greece on the Mediterranean coast, first the rise of the Macedonian dynasty, and then the rise of the Roman dynasty. After the Romans unified Italy, they expanded westward and met another powerful country, Carthage. The two countries fought for 23 years from 264 BC to 221 BC. This was the famous "First Punic War" in history, and the Romans won. The war lasted for another four years starting in 218 BC. This was the "Second Punic War." This time Carthage hired a slave-born military strategist, Hannibal, to capture more than 50,000 Romans in one fell swoop. The two powerful countries along the Mediterranean coast have been fighting for years, with both sides winning and losing. Syracuse was a small city-state sandwiched between the two powerful countries of Cara and Luo. In this long-term war situation, it often abandoned the weak and attached itself to the strong with the victory or defeat of the two powerful countries, which was erratic. Archimedes was very worried about this diplomatic strategy and warned the king many times not to get into trouble. But the current king is no longer the good friend of Archimedes, Hanero. He is young and ignorant, yet headstrong. When the 'Second Punic War' broke out in 216 BC, seeing that the Carthaginians were about to defeat the Romans, the king quickly broke with the Romans and formed an alliance with the Carthaginians. This move is very annoying. Now that the Romans had won another battle, they took revenge and attacked this small city-state from both sea and land. The king was so frightened that he lost his mind. When he saw Archimedes coming in from the outside, he rushed forward to greet him. He wanted to kneel down to him immediately and said: 'Ah, dear Archimedes, you are the smartest man. The late king said it when he was alive. You can move the earth. ’
The story about Archimedes pushing the earth happened when he was studying abroad in Alexandria. At that time, he was inspired by the booms used by Egyptian farmers to lift water and the crowbars used by slaves to pry stones. He discovered that a lever could be used to save labor. He also found that the longer the distance between the hand and the fulcrum. , the more effort you will save. From this, he proposed such a theorem: The relationship between moment arm and force (weight) is inversely proportional. This is the principle of leverage. Using our current expression, it is: weight × heavy arm = force × moment arm. For this reason, he wrote to the then king Highnillo and said: "I can move anything of any weight without any effort; as long as you give me a fulcrum and a lever long enough, I can even move the earth." Can be pushed. ’ But now this little king doesn’t know what science is. He only knows how to use Archimedes’ magical power to save his driver when disaster strikes.
But the Roman army was too powerful. When they fought, they formed a square formation. The soldiers in front and on both sides protected their bodies with shields, and the soldiers in the middle held their shields over their heads. When the war drums sounded, each square formation was like a modern tank, marching towards the enemy camp. Push forward step by step, no matter how many arrows you shoot at, there will be no damage at all. The Roman army also had particularly strict military discipline. Anyone found to have escaped from battle would be executed immediately. Soldiers would be promoted for meritorious service. A grand triumphal ceremony would be held when the commander-in-chief returned to Rome after victory. This army dominates the Mediterranean and is invincible. How could it care about a small Syracuse?
Besides, there are old grudges and new grudges, and I have long wanted to have a thorough settlement. At this time, four army legions commanded by the Roman consul Marcellus had advanced to the northwest of the ancient city of Syracuse. Now the drums are beating outside the city, and the sound of killing is shaking the sky. At this critical moment, although Archimedes was very dissatisfied with the disaster caused by the king's short-sightedness, the work was done and the country was the most important thing. He glanced at the dull hall, twirled his silver beard and said: 'If we rely solely on In terms of military strength, we are by no means a match for the Romans. Now if a new type of weapon can be made, it may be possible to defend the city and wait for reinforcements. ’ When the king heard this, he immediately turned from worry to joy and said: ‘The late king said long ago when he was still alive that everyone must believe whatever you say. You will have full command of this guard battle. ’
Two days later, at dawn, the Roman commander Marcellus commanded his tight and neat phalanx to attack the moat. Today, armored cavalry were also prepared on both sides of the phalanx, and the strong soldiers in the phalanx carried ladders on their shoulders. Before setting off, Marcellus uttered wild words: "Beat Syracuse and have lunch in the city." ’ Amidst the shouts of killing, the phalanx slowly wriggled forward. As usual, arrows should have been fired at the top of the city. But today there was no one on the city wall quietly. Perhaps the Syracusans were exhausted after several days of fierce fighting. The Romans were wondering when a faint creaking sound came from the city, and then large and small stones flew out from the top of the city. At first, they were as big as bowls and fists, and then they became bigger and bigger, almost like pots and pans. , pouring down like a mountain torrent. The stone fell in the enemy's formation, and the soldiers quickly raised their shields to protect themselves. Unexpectedly, the stone was so heavy and so fast that all the people with their shields were smashed into a ball of flesh. The Romans gradually couldn't hold on anymore and rolled and crawled to the ground to escape. At this time, another dense stream of sharp arrows was fired from the city of Syracuse. The Romans had no shields or armor to resist them. The sharp arrows pierced their backs, making the heavens and the earth cry out. It was so miserable.
What secret weapon did Archimedes create that allowed the Romans to return in defeat? It turned out that he created some extremely large crossbows-stone machines. It was impossible for anyone to pull such a big bow, so he used the principle of leverage. As long as the crank handle of the rotating shaft on the crossbow is pulled hard, the beef tendons connected to the crank handle will tighten the thick bow string composed of many beef tendons. When it is pulled to the tightest point, and then suddenly released, the bow string will drive the stone-carrying device. If the stone is thrown high out of the city, it will land more than 1,000 meters away. It turns out that the principle of leverage is not simply to use a straight stick to pry things. For example, take the windlass on a water well. Its fulcrum is the axis of the windlass, its heavy arm is the radius of the windlass, and its moment arm is the rocking handle. The rocking handle must be longer than the radius of the windlass. It saves effort when pumping water. Archimedes' stone machine also used this principle. How did the Romans know that there were so many new gadgets in the ancient city of Syracuse?
Not long after Marcellus was defeated, Navy Commander Claudius also sent someone to send a battle report. It turned out that when the army attacked the city from the northwest, the Roman navy also launched an offensive from the southeast sea. The Roman navy was not very powerful at first. Later, it invented a side hook mounted on the ship. When it encountered an enemy ship, it hooked the enemy ship. The soldiers then jumped on the enemy ship, turning the sea battle into a land battle, and gaining a certain advantage. Today, in order to deal with Syracuse, Claudius specially covered the warships with a layer of iron armor, prepared ladders, and ordered the soldiers to only move forward and not retreat. Strangely, the city head of Syracuse was extremely quiet that day. There was not a single soldier visible behind the wall, but a few wooden frames standing on the city head could be seen from a distance. When the Roman warships arrived at the city, the soldiers were holding ladders and were about to build them up the wall. Suddenly, iron chains hung from the wooden frames. There were iron hooks and claws on the chain heads, which hooked the Roman navy's ships. Warship. No matter how hard the sailors rowed, it was all in vain, and the warship could no longer move even half a step. They chopped it with knives and burned it with fire, but the big iron chain was not damaged at all. Just when there was panic on the ship. I saw the wooden wheel on the big wooden frame turning again, and then the iron chain became tighter and tighter, and the boat was gradually lifted out of the water. As the hull tilted, the soldiers fell into the sea one after another, and the mast was broken. After the hull is lifted into the air, the large wooden frame will still rotate left and right.
So the warships swayed in the air like swings, and then some were thrown onto the city wall or rocks and turned into pieces; some were hoisted over the city wall and became the trophies of the Syracusans. At this time, the city of Syracuse was still quiet, no one was shooting arrows, and no one was shouting. It seemed like an empty city, with only a few monster-like wooden frames, and from time to time, big hooks were stretched out to catch ships. Warship. When the Romans looked at the rattling monster, they were so frightened that their whole bodies trembled and their hands and legs became weak. They could only hear cries on the sea and calls for help after falling into the water and hitting rocks. Claudius said in the battle report: "We can't see the enemy at all, it's like fighting a barrel." ’ These ‘monsters’ of Archimedes also used the lever principle and added pulleys.
After this battle, the Romans lost many troops and many weapons and ships, but they did not even see Archimedes.
Example demonstration
There are basically three types of lever principles. Examples of the first type of levers are balances, scissors, pliers, etc., and examples of the second type of levers are corkscrews and walnuts. Clamps, the third type of levers such as hammers, tweezers, etc.