전세계 3%만 풀 수 있는 문제?…“정답은 셋 중 하나

[진세]

[매경닷컴 MK스포츠 김수미 기자] 보는 이들의 도전심을 자극하는 ‘전 세계 3%만 풀 수 있는 문제’가 화제다.

최근 온라인 상에는 `전세계 3%만 풀 수 있는 문제`라는 제목으로 한 과학 문제가 등장해 누리꾼들의 머릿속을 복잡하게 만들었다.

문제는 커다란 그릇 속에 얼음이 그림과 같이 떠 있다. 얼음이 녹으면 그릇 속에 물은 어떻게 될까를 묻는 문제이다. 보기는 ‘늘어난다, 줄어든다, 그대로이다’의 세가지다.

이어 문제를 푼 정답자는 “부력의 가장 기본적인 개념이니 어렵게 생각할 필요 없다. 왼쪽의 얼음에서 위에 떠 있는 부분은 물에 대한 얼음의 비중인 약 0.9%의 남는 부분, 즉 얼음의 10%정도이다. 물이 얼음으로 바뀌면 부피가 약 1.1배 증가하고, 그 증가부분이 위에 뜬다고 생각하면 된다”라고 설명했다.

그는 이어 “마찬가지로 비중이 1인 물체가 있다면 그 물체는 가라앉기 직전 정확히 수면에 걸려서 떠 있을 거다”고 말했다.

이어 정답자는 “만약 이 얼음이 녹게되면 오른쪽의 파란색 물이 되면서 부피가 다시 9/10으로 줄게 된다. 그러면 물에 떠 있는 부분에서의 10%와 가라앉은 부분에서의 10%가 없어진다. 그 없어지는 부분이 얼음 상태에서의 떠 있는 부피와 똑같은 양이다”면서 “이게 아르키메데스의 부력의 원리다”라고 덧붙였다. 부력F=q(물체가 잠긴 액체의 밀도)xV(물체로 인해 밀려난 액체의 부피)xg(중력가속도)라는 설명이다.

아르키메데스(Archimedes)가 발견한 원리에 의하면 어떤 물체이건 물에 잠기게 되면 그 물체의 부피에 상당하는 물의 무게만큼 부력을 받는다.

이 게시물은 실제로 이 설명이 답이 아닌데다, 아무리 읽어도 답이 무엇인지 모르는 장황한 설명으로 더욱 화제가 됐다. 해당 내용의 답을 주장한 사람과 대부분의 누리꾼들은 3번을 정답으로 꼽았지만 물의 온도, 대기중의 온도, 기압, 얼음의 크기, 그릇의 모양,크기 등에 따라 실제로는 많은 오차가 발생할 수 있기 때문에 애초에 문제 자체가 오류이다.

누리꾼들은 다양한 의견들을 내놓으면 전 세계 3%만이 풀 수 있는 문제에 의욕을 불태웠다.

아르키메데스의 원리 [─原理, Archimedes' principle]

부력의 원리라고도 한다. 유체(기체나 액체) 속에 정지해 있는 물체는 중력과 반대방향으로 부력, 즉 물체를 위로 뜨게 만드는 힘을 받는다. 부력은 물체 주위의 유체가 물체에 미치는 압력의 합이라고 할 수 있다. 이 힘의 크기는 물체를 그 유체로 바꾸었을 때 작용하는 중력의 크기와 같다. 다시 말해, 물체의 부피에 해당하는 유체의 무게와 같다.

이 원리는 BC 220년경 아르키메데스가 발견했다고 하여 그의 이름이 붙었다. 그는 물 속에 있는 물체가 실제의 무게보다 가볍게 느껴지는 현상을 설명하는 데 이 원리를 이용했다. 이 원리는 물질의 비중을 측정하는 기초가 되며 옛날부터 뱃짐 등의 복잡한 형태를 가진 물체의 부피를 측정하는 데 사용하였다. 복잡하게 생긴 물체의 부피는 직접 측정하기 힘들다. 대신 물체를 물과 같이 비중을 아는 액체 속에 담그고 무게를 달아, 담그기 전보다 얼마나 무게가 줄어들었는지를 측정하면 부피를 알 수 있다.

전설에 의하면 아르키메데스는 시라쿠사의 왕 히에론의 명에 따라 왕관이 순금으로 만든 것인지를 조사하게 된다. 그는 고심하던 중에 우연히 목욕탕에서 물에 들어가면 몸의 부피만큼 물이 넘쳐 흐른다는 것을 깨닫고 물체의 부피와 무게와의 관계를 밝혔다고 한다. 은이나 구리 등의 물질은 금보다 밀도가 작기 때문에 같은 질량의 금보다 그 부피가 더 크다. 따라서 은이나 구리 등을 섞어서 왕관을 만들었다면 같은 질량의 금으로 만든 왕관보다 그 부피가 더 클 것이다. 아르키메데스는 왕관과 또 그것과 같은 질량의 금을 따로따로 물 속에 담그고 넘쳐 흘러나온 물의 부피를 측정하였다. 그리고 왕관을 넣은 쪽에서 흘러나온 물이 더 많다는 것을 근거로 왕관이 순금으로 만들어지지 않았다는 것을 알아내었다. 

 

Archimedes' principle

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Archimedes' principle relates buoyancy to displacement. It is named after its discoverer, Archimedes of Syracuse.^[1]

Contents

 [hide] 

• 1 Principle
• 2 Refinements
• 3 Formula
• 4 See also
• 5 References

[edit] Principle

Archimedes' treatise On floating bodies, proposition 5, states that

Any floating object displaces its own weight of fluid.

—Archimedes of Syracuse^[2]

For more general objects, floating and sunken, and in gases as well as liquids (i.e. a fluid), Archimedes' principle may be stated thus in terms of forces:

Any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object.

— Archimedes of Syracuse

with the clarifications that for a sunken object the volume of displaced fluid is the volume of the object, and for a floating object on a liquid, the weight of the displaced liquid is the weight of the object.

In short, buoyancy = weight of displaced fluid.

[edit] Refinements

Archimedes' principle does not consider the surface tension (capillarity) acting on the body.^[3]

[edit] Formula

The weight of the displaced fluid is directly proportional to the volume of the displaced fluid (if the surrounding fluid is of uniform density). In simple terms, the principle states that the buoyant force on an object is going to be equal to the weight of the fluid displaced by the object, or the density of the fluid multiplied by the submerged volume times the gravitational constant, g. Thus, among completely submerged objects with equal masses, objects with greater volume have greater buoyancy.

Suppose a rock's weight is measured as 10 newtons when suspended by a string in a vacuum with gravity acting upon it. Suppose that when the rock is lowered into water, it displaces water of weight 3 newtons. The force it then exerts on the string from which it hangs would be 10 newtons minus the 3 newtons of buoyant force: 10 − 3 = 7 newtons. Buoyancy reduces the apparent weight of objects that have sunk completely to the sea floor. It is generally easier to lift an object up through the water than it is to pull it out of the water.

Assuming Archimedes' principle to be reformulated as follows,

then inserted into the quotient of weights, which has been expanded by the mutual volume

yields the formula below. The density of the immersed object relative to the density of the fluid can easily be calculated without measuring any volumes:

(This formula is used for example in describing the measuring principle of a dasymeter and of hydrostatic weighing.)

Example: If you drop wood into water, buoyancy will keep it afloat.

Example: A helium balloon in a moving car. In increasing speed or driving a curve, the air moves in the opposite direction of the car's acceleration. The balloon however, is pushed due to buoyancy "out of the way" by the air, and will actually drift in the same direction as the car's acceleration. "When an object is immersed in a liquid the liquid exerts an upward force which is known as buoyant force an it is equal to the weight of the object"

Eureka (word)

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For other uses, see Eureka (disambiguation).

"Eureka" (pronounced /jʊˈriːkə/) is an interjection used to celebrate a discovery, a transliteration of a word attributed to Archimedes.

Contents

 [hide] 

• 1 Etymology
• 2 Archimedes
• 3 Names and mottos
• 4 Mathematics
• 5 See also
• 6 References

[edit] Etymology

The word comes from ancient Greek εὕρηκα heúrēka "I have found (it)", which is the 1st person singular perfect indicative active of the verb heuriskō "I find".^[1] The reconstructed Ancient Greek pronunciation is [hěu̯rɛːka], while the Modern Greek pronunciation is [ˈevrika].

The accent of the English word is on the second syllable, following Latin accent rules, which require that a penult (next-to-last syllable) must be accented if it has a long vowel. In the Greek pronunciation, the first syllable has a high pitch accent, because the Ancient Greek rules of accent do not force accent to the penult unless the ultima (last syllable) has a long vowel. The long vowels in the first two syllables would sound like a double stress to English ears (as in the phrase Maltese cat).

The initial /h/ is dropped in some European languages, including English, but preserved in others, such as Finnish and German: Heureka.

[edit] Archimedes

This exclamation is most famously attributed to the ancient Greek scholar Archimedes. He reportedly proclaimed "Eureka!" when he stepped into a bath and noticed that the water level rose—he suddenly understood that the volume of water displaced must be equal to the volume of the part of his body he had submerged. This relation is known as Archimedes' principle. He then realized that the volume of irregular objects could be measured with precision, a previously intractable problem. He is said to have been so eager to share his discovery that he leapt out of his bathtub and ran through the streets of Syracuse naked.

Archimedes' insight led to the solution of a problem posed by Hiero of Syracuse, on how to assess the purity of an irregular golden votive crown; he had given his goldsmith the pure gold to be used, and correctly suspected he had been cheated, by the goldsmith removing gold and adding the same weight of silver. Equipment for weighing objects already existed, and now that Archimedes could also measure volume, their ratio would give the object's density, an important indicator of purity.

This story first appeared in written form in Vitruvius's books of architecture, two centuries after it supposedly took place.^[2] Some scholars have doubted the accuracy of this tale, saying among other things that the method would have required precise measurements that would have been difficult to make at the time.^[3] For the problem posed to Archimedes, though, there is a simple method which requires no precision equipment: balance the crown against pure gold in air, and then submerge the scale with crown and gold in water to see if they still balance.^[4]

[edit] Names and mottos

The expression‎ is also quoted as the state motto of California, referring to the momentous discovery of gold near Sutter's Mill in 1848. The California State Seal has included the word "eureka" since its original design by Robert S. Garnett in 1849; the official text from that time describing the seal states that this word's meaning applies "either to the principle involved in the admission of the State or the success of the miner at work". In 1957, the state legislature attempted to make "In God We Trust" the state motto, but this attempt did not succeed, and "Eureka" became the official motto in 1963.^[5]

The city of Eureka, California, founded in 1850, uses the California State Seal as its official seal. Eureka is a considerable distance from Sutter's Mill, but was the jumping off point of a smaller gold rush in Trinity County, California in 1850. It is the largest of at least eleven remaining US cities and towns named for the exclamation, "eureka!". As a result of the extensive use of the exclamation dating from 1849, there were nearly 40 locales so named by the 1880s in a nation that had none in the 1840s.^[6] Many places, works of culture, and other objects have since been named "Eureka"; see Eureka (disambiguation) for a list.

"Eureka" was also associated with a gold rush in Ballarat, Victoria, Australia. The Eureka Stockade was a revolt in 1854 by gold miners against unjust mining license fees and a brutal administration supervising the miners. The rebellion demonstrated the refusal of the workers to be dominated by unfair government and laws. The Eureka Stockade has often been referred to as the 'birth of democracy' in Australia.

[edit] Mathematics

Another mathematician, Carl Friedrich Gauss, echoed Archimedes when in 1796 he wrote in his diary, "ΕΥΡΗΚΑ! num = Δ + Δ + Δ", referring to his discovery that any positive integer could be expressed as the sum of at most three triangular numbers.^[7] This result is now known as Gauss' Eureka theorem^[8] and is a special case of what later became known as the Fermat polygonal number theorem.