Kayan Dokar Archimedes
Shin ka taɓa ganin jirgin ruwa? Idan ba ka taɓa ganinsa da idonka ba, da fatan ka taɓa ganinsa a talabijin ko intanet. Jirgin ruwa mai girman gaske ba ya nutsewa, yayin da ƙaramin dutse mai sauƙi zai iya nutsewa. Me ya sa haka? Amsar tana da sauƙi idan ka fahimci manufofin buoyancy da ƙa'idar Archimedes.
A rayuwar yau da kullum, za mu ga cewa abubuwan da aka sanya a cikin ruwa, kamar dutse, suna da ƙarancin nauyi fiye da lokacin da abin ba ya cikin ruwan. Kuna iya samun matsala wajen ɗaga dutse daga saman ƙasa, amma dutsen ɗaya yana da sauƙin ɗagawa daga ƙasan tafkin. Wannan ya faru ne saboda ƙarfin buɗaɗɗen ruwa. Buɗaɗɗen ruwa yana faruwa ne saboda bambancin matsin lamba a zurfin daban-daban. Matsin ruwa yana ƙaruwa da zurfi, zurfin ruwan yana ƙaruwa, matsin lamba na ruwa yana ƙaruwa. Lokacin da aka sanya abu a cikin ruwa, za a sami bambanci a matsin lamba tsakanin ruwan da ke sama da abin da kuma ruwan da ke ƙasa da abin. Ruwan da ke ƙasan abin yana da matsin lamba mafi girma fiye da ruwan da ke saman abin.
Buɗewa
A cikin hoton, an ga wani abu yana shawagi a cikin ruwa. Ruwan da ke ƙarƙashin abin yana da matsi mafi girma fiye da ruwan da ke saman abin. Wannan saboda ruwan da ke ƙarƙashin abin yana da zurfin da ya fi na saman abin (h)2 > h1).
F2 = ƙarfin da ruwan da ke ƙasan abin ke fitarwa, F1 = ƙarfin da ruwan da ke saman abin ke fitarwa, A = yankin saman abin
Bambanci tsakanin F2 da kuma F1 shine jimlar ƙarfin da ruwan ke yi akan wani abu, wanda muka sani da ƙarfin buwaya. Girman ƙarfin buwaya shine:
F mai ƙarfi = F2 − F1
F buoyancy = (ρ gh2 A) − (ρ gh1 A)
F mai ƙarfi = ρ g A ( h2 - h1)
F buoyancy = ρ g A h
F mai ƙarfi = ρ g V
Bayani:
ρ = yawan ruwa
g = hanzari saboda nauyi
V = girman abin da ke cikin ruwan
don haka za a iya rubuta lissafin da ya nuna girman ƙarfin buoyant (F buoyancy) a sama kamar haka:
F buoyancy = ρF g V → m = ρ V
F mai ƙarfi = mF g
F mai ƙarfi = wF
mF g = wF = nauyin ruwa wanda yake da girman daidai da girman abin da ke cikin ruwa.
Ka'idar Archimedes
Bisa ga lissafin da ke sama, za mu iya cewa ƙarfin buoyant yana daidai da nauyin ruwan da aka kora, girman ruwan da aka kora daidai yake da girman abin da aka nutsar a cikin ruwan.
Idan wani abu ya nutse a cikin ruwa ya yi iyo, sai wani ɓangare na abin ya nutse kawai, to girman ruwan da aka kora zai yi daidai da girman ɓangaren abin da aka nutse a cikin ruwan. Ko da kuwa siffar ko girman abin ne, abu ɗaya zai faru da kowa. Wannan aikin Archimedes ne (287-212 BC), wanda yanzu ake kira Archimedes' Principle.
Ka'idar Archimedes ta bayyana cewa:
Idan aka nutsar da wani abu gaba ɗaya ko kuma wani ɓangare na shi a cikin ruwa, ruwan zai yi amfani da ƙarfin sama (ƙarfin buoyant) akan abin, inda girman ƙarfin sama (ƙarfin buoyant) yayi daidai da nauyin ruwan da aka motsa.
Labarin Archimedes
Archimedes, wanda ya rayu tsakanin 287 da 212 kafin haihuwar Annabi Isa (AS), Sarki Hieron na Biyu ne ya ba shi umarnin ya binciki ko an yi kambin da aka yi wa sarki da zinariya tsantsa. Da farko Archimedes ya rikice game da ko an yi kambin da zinariya tsantsa ko kuma yana ɗauke da wasu ƙarfe. Matsalar ita ce kambin ba shi da tsari kuma ba za a iya lalata shi ba don a tantance ko zinariya tsantsa ce.
Manufar da ke bayan tantance ko an yi kambin sarki da zinariya tsantsa ita ce a fara tantance takamaiman nauyin kambin sannan a kwatanta shi da takamaiman nauyin zinariya. Idan kambin ya kasance da zinariya tsantsa, to takamaiman nauyin kambin = takamaiman nauyin zinariya.
Takamaiman nauyin abu shine rabon nauyin abu a cikin iska da nauyin ruwa mai girma iri ɗaya da abin. A lissafi, an rubuta:
Yaya za a tantance nauyin ruwa wanda yake da girma iri ɗaya da girman abu?
A cewar Archimedes, nauyin ruwa mai girman daidai da girman abin = girman ƙarfin tururi lokacin da abin ya nutse (dukan abin yana nutsewa cikin ruwa). Wannan daidai yake da nauyin abin da aka rasa lokacin da aka auna shi a cikin ruwa. Saboda haka:
Domin tantance takamaiman nauyin kambin, da farko ana auna kambin a cikin iska (Nauyin Kambin a Sama). Bayan haka, ana saka kambin a cikin ruwa sannan a sake auna shi don samun nauyin kambin da ya ɓace. Don haka:
Da zarar an sami takamaiman nauyin kambin, sai a kwatanta shi da takamaiman nauyin zinariya. Takamaiman nauyin zinariya shine 19,3. Idan takamaiman nauyin kambin ya yi daidai da takamaiman nauyin zinariya, to an yi kambin da zinariya tsantsa. Duk da haka, idan ba a yi kambin da zinariya tsantsa ba, to takamaiman nauyin kambin ba zai yi daidai da takamaiman nauyin zinariya ba.
Me Yasa Jiragen Ruwa Ba Su Nutse Ba?
Idan yawan abu ya yi ƙasa da yawan ruwa, to abin zai yi iyo. A gefe guda kuma, idan yawan abu ya fi yawan ruwa, abin zai nutse. Yawancin jiragen ruwa an yi su ne da ƙarfe da ƙarfe. Yawan ƙarfe da ƙarfe = 7,8 x 103 kg / m3 yayin da yawan ruwa = 1,00 x 103 kg / m3Da alama yawan ƙarfe da ƙarfe ya fi yawan ruwa. A wannan yanayin, takamaiman nauyin ƙarfe da ƙarfe = 7,8. Ya kamata jiragen ruwa da aka yi da ƙarfe da ƙarfe su nutse. Me yasa jiragen ruwa ba sa nutsewa? Jimillar yawan jirgin ya fi na ruwa ko ruwan teku ƙanƙanta.
Misalan tambayoyi kan dokar Archimedes
Misali tambaya ta 1:
Dutse mai nauyin kilogiram 40 yana ƙasan tafkin. Idan girman dutsen = mita 0,23 nawa ne ƙarfin mafi ƙarancin abin da ake buƙata don ɗaga dutsen?
Tattaunawa
An san cewa:
Nauyin dutse (m) = 40 Kg
Girman dutse (V) = 0,02 m3
Yawan ruwa = 1000 Kg/m23
Saurin gudu saboda nauyi (g) = 10 m/s2
An tambaya: Mafi ƙarancin F
Amsa:
F mai ƙarfi = w F
F buoyancy = m F g → m = pV
F buoyancy = ρ F g V
F mai ƙarfi = (1000 kg/m2)3 )(10 m/s2)(mita 0,023)
F mai ƙarfi = 200 kg m/s2
F mai ƙarfi = 200 N
Nauyin dutse (w) = mg
Nauyin dutse = (kilogiram 40)(mita 10/s)2)
Nauyin dutse = 400 kg m/s2
Nauyin dutse = 400 N
Mafi ƙarancin ƙarfi da ake buƙata don ɗaga dutse:
Nauyin dutse – Ƙarfin busawa = 400 N – 200 N = 200 Newton.
Misali tambaya ta 2:
Nauyin abu a cikin iska = 5000 kg m/s2 kuma nauyin abin a cikin ruwa = 4000 kg m/s2Idan yawan abu = 2000 kg/m3 Menene nauyin da girman abin? g = 10 m/s2
Tattaunawa
Saurin gudu saboda nauyi (g) = 10 m/s2
Yawan abu = 2000 kg/m3
Yawan ruwa = 1000 kg/m23
Nauyin abu a cikin iska = 5000 kg m/s2
Nauyin abu a cikin ruwa = 4000 kg m/s2
Ƙarfin Buoyant (F buoyant) = Nauyin abu a cikin iska - Nauyin abu a cikin ruwa
F mai ƙarfi = 5000 kg m/s2 – 4000 kg m/s2
F mai ƙarfi = 1000 kg m/s2
F buoyancy = Nauyin ruwan da aka kora
F mai ƙarfi = (tarin ruwa)(g)
F buoyancy = (yawan ruwan da aka kora)(yawan ruwa)(g)
Yawan ruwan da aka kora = Yawan abin da aka nutsar a cikin ruwa 😉
Don haka girman abin = 0,1 m3
Nauyin abu = ?
ρ = m / V
m = ρ V
m = ( 2000 kg / m3 )( 0,1 m3 )
m = 200 Kg
Nauyin abu = 200 kg
Misali tambaya ta 3:
Wane girman helium ake buƙata idan balan-balan dole ne ya ɗaga kaya na kilogiram 500?
Tattaunawa
Yawan helium = 0,1786 kg/m23
Yawan iska = 1,293 kg/m23
Ƙarfin buoyant = Nauyin iska da aka motsa = Nauyin kaya + Nauyin helium
Ƙarfin buoyant = Nauyin kaya + nauyin helium
Ƙarfin ƙaruwa = (tarin kaya)(g) + (tarin helium)(g)
Ƙarfin ƙaruwa = (tarin kaya + nauyin helium) g —- lissafi na 1
Ƙarfin tururin iska = Nauyin iskar da aka kora
Ƙarfin hayaniya = (tarin iskar da aka motsa)(g) —- lissafi na 2
Mun haɗa lissafi na 1 da lissafi na 2:
(tarin kaya + nauyin helium)(g) = (tarin iskar da aka cire)(g)
nauyin kaya + nauyin helium = nauyin iska da aka kora
500 kg + (ρ helium)(V helium) = (ρ iska)( V iska)
500 kg = (ρ iska)(V iska) – (ρ helium)(V helium)
Ƙarar iskar da aka fitar (V iska) = Ƙarar helium da ke cika balan-balan (V helium)
500 kg = (ρ iska - ρ helium) (V)
Wannan shine ƙaramin girman helium da ake buƙata don ɗaga nauyi a saman Duniya. Domin barin balan-balan ya yi iyo sama da haka, ana buƙatar ƙara girman helium. Wannan saboda yawan iska yana raguwa tare da tsayi.
