Ka'idar Bernoulli da Kayan Daidaito
Idan muka hau babur da sauri, tufafin da muke sawa suna ƙara ƙarfi a baya. Idan ba za ku iya hawa babur ba tukuna, ku kula da iyayenku ko abokanku waɗanda ke hawa babur. Bayan tufafinsu yawanci yana ƙara ƙarfi a baya lokacin da babur ɗin ke tafiya da sauri. Wani lokaci idan iska ta busa da ƙarfi, ƙofar gidan na iya rufewa da kanta. Ko da yake iska tana busawa a wajen gidan, yayin da ƙofar ke cikin gidan.
Ana iya bayyana wannan ta amfani da ƙa'idar Bernoulli. Daniel Bernoulli (1700–1782) ya gano wata ƙa'ida da za a iya amfani da ita don bayyana wasu daga cikin abubuwan da ke sama.
Ka'idar Bernoulli
Ka'idar Bernoulli ta bayyana cewa inda saurin kwararar ruwa yake da yawa, matsin lamba na ruwa yana da ƙasa. Akasin haka, idan saurin kwararar ruwa ya yi ƙasa, matsin lamba na ƙaruwa. Lokacin da babur ke tafiya da sauri, saurin iska a gaba da gefen jikinka yana da yawa. Don haka, matsin lamba na iska yana raguwa. Bayan jikinka yana toshewa ta gaban jikinka, don haka saurin iska a bayan jikinka baya canzawa zuwa sama (a bayan jikinka). Sakamakon haka, matsin lamba na iska a bayan jikinka yana ƙaruwa. Saboda akwai bambanci a matsin lamba na iska, inda matsin lamba na iska a bayan jikinka ya fi girma, iska tana tura rigarka baya, tana sa ta yi kama da an hura ta a baya.
Yaya batun ƙofar da ke rufewa da kanta lokacin da iska ke busawa da ƙarfi a waje? Iskar waje tana motsawa da sauri fiye da iskar da ke ciki. Sakamakon haka, matsin lamba a waje ya fi ƙasa da matsin lamba a ciki. Saboda wannan bambancin matsin lamba, inda matsin lamba a ciki ya fi girma, ƙofar tana tura waje. A wata ma'anar, ƙofar tana motsawa daga wurin matsin lamba mafi girma zuwa wurin matsin lamba mafi ƙasa.
Daidaiton Bernoulli
A baya, mun koyi game da ƙa'idar Bernoulli. Bernoulli kuma ya haɓaka wannan ƙa'idar ta hanyar adadi. Domin samo lissafin Bernoulli, muna ɗaukar kwararar ruwa mai laminar, rashin matsewa, da ƙarancin ɗanko, wanda za a iya watsi da shi.
A cikin tattaunawar Daidaito na Ci gaba, mun koyi cewa saurin kwararar ruwa na iya bambanta dangane da yankin giciye na bututun kwarara. Dangane da ƙa'idar Bernoulli da aka bayyana a sama, matsin lamba na ruwa na iya bambanta dangane da saurin kwararar ruwa. Matsin ruwa kuma na iya bambanta dangane da tsayin ruwan. Ana iya samun alaƙar da ke tsakanin matsin lamba, saurin kwarara, da tsayin kwarara daga lissafin Bernoulli.
Lissafin Bernoulli yana da matuƙar muhimmanci domin ana iya amfani da shi wajen nazarin tashi daga jirgin sama, tashoshin wutar lantarki na ruwa, tsarin bututu, da sauransu. Domin mu sa lissafin Bernoulli da za mu samu ya yi aiki gabaɗaya, za mu ɗauka cewa ruwan yana gudana ta cikin bututun kwarara mai yankunan giciye marasa daidaito da tsayi daban-daban (duba hoton da ke ƙasa). Don samo lissafin Bernoulli, za mu yi amfani da ka'idar aiki da makamashi ga ruwan da ke yankin bututun kwarara. Na gaba, za mu ƙididdige adadin ruwa da aikin da aka yi don motsa ruwan.
Launin da ba a iya gani a cikin bututun kwararar ruwa a cikin hoton yana nuna kwararar ruwa yayin da launin fari yana nuna babu ruwa.
Ruwan da ke cikin yankin giciye 1 (gefen hagu) yana gudana tazara L 1 kuma yana tilasta ruwan da ke cikin sashe 2 (gefen dama) ya motsa tazara L 2. Saboda yankin giciye 2 a gefen dama ya fi ƙanƙanta, ƙimar kwararar ruwa a gefen dama na bututun kwarara ya fi girma (Ku tuna da daidaiton ci gaba). Wannan yana haifar da bambancin matsin lamba tsakanin sashe 2 (gefen dama na bututun kwarara) da sashe 1 (gefen hagu na bututun kwarara) - Ku tuna da ƙa'idar Bernoulli. Ruwan da ke gefen hagu na sashe 1 yana motsa matsin lamba P 1 akan ruwan da ke gefen dama kuma yana aiki da:
Sannan za a iya rubuta lissafin W 1 kamar haka:
W 1 = p 1 A 1 L 1
A giciye sashe na 2 (gefen dama na bututun kwarara), aikin da aka yi akan ruwan shine:
W 2 = − p 2 A 2 L 2
Alamar mara kyau tana nuna cewa ƙarfin da aka yi amfani da shi yana akasin alkiblar motsi. Don haka ruwan yana aiki a gefen dama na sashe na 2. Bugu da ƙari, ƙarfin nauyi shi ma yana aiki akan ruwan. A cikin yanayin da ke sama, ana motsa wani taro na ruwa daga sashe na 1 har zuwa L 1 zuwa sashe na 2 har zuwa L 2 , inda girman ruwa a sashe na 1 (A 1 L 1 ) = girman ruwa a sashe na 2 (A 2 L 2 ). Aikin da nauyi ke yi shine:
W 3 = − mg (h 2 − h 1 )
W 3 = − mg 2 + mg 1 )
W 3 = mg 1 − mg 2
Alamar mara kyau ta faru ne saboda ruwan da ke kwarara sama, akasin alkiblar nauyi. Don haka, jimlar aikin da aka yi akan ruwan, kamar yadda aka nuna a cikin hoton da ke sama, shine:
W = W 1 + W 2 + W 3
W = P 1 A 1 L 1 - P 2 A 2 L 2 + mg 1 - mg 2
Ka'idar aiki-makamashi ta bayyana cewa jimlar aikin da aka yi akan wani tsari daidai yake da canjin kuzarin motsi. Don haka, za mu iya maye gurbin Work (W) da canjin kuzarin motsi (EK 2 - EK 1 ).
Za mu iya sake rubuta lissafin da ke sama kamar haka:
W = P 1 A 1 L 1 - P 2 A 2 L 2 + mg 1 - mg 2
EK 2 - EK 1 = P 1 A 1 L 1 - P 2 A 2 L 2 + mgh 1 - mgh 2
1⁄2 mv 2 2 ‐ 1⁄2 mv 1 2 = P 1 A 1 L 1 ‐ P 2 A 2 L 2 + mg 1 ‐ mg 2
Nauyin ruwa yana gudana tazara L 1 a giciye sashe A 1 = nauyin ruwa yana gudana tazara L 2 (sashe na A 2 ). Nauyin ruwa, kamar m, yana da girman A 1 L 1 da A 2 L 2 inda A 1 L 1 = A 2 L 2 (L 2 ya fi L 1 tsayi ).
Yanzu muna maye gurbin, ko kuma mu maye gurbin m a cikin lissafin da ke sama da m = ρ AL:
Wannan shine lissafin Bernoulli. Mun samo lissafin Bernoulli bisa ga ƙa'idar aiki da makamashi, don haka wani nau'i ne na Dokar Kare Makamashi.
Bayani:
P = Matsi
ρ = Yawan ruwa
v = Gudun kwararar ruwa
g = Hanzari saboda nauyi
h = Tsayin bututun/bututun kwarara daga saman ƙasa
Gefen hagu da dama na lissafin Bernoulli da ke sama na iya nufin kowane maki biyu a kan bututun kwarara don haka za mu iya sake rubuta lissafin da ke sama kamar haka:
Wannan lissafin ya nuna cewa jimillar adadin da ke cikin lissafin yana da ƙima iri ɗaya a cikin bututun kwarara.
Yanzu bari mu sake duba lissafin Bernoulli don wasu lokuta.
Daidaiton Bernoulli don Ruwa a Hutu
Misali na musamman na lissafin Bernoulli shine na ruwa a lokacin hutawa (ruwa mai tsayayye). Idan ruwa yana hutawa, ba shi da saurin gudu. Don haka, v 1 = v 2 = 0. Idan ruwa yana hutawa, ana iya tsara lissafin Bernoulli kamar haka:
Idan h 2 - h 1 = h, to wannan lissafin za a iya rubuta shi kamar haka:
p 1 − p 2 = ρ g (h 2 − h 1 )
p 1 − p 2 = ρ gh
Daidaiton Bernoulli na bututun kwarara ko bututu masu tsayi ɗaya
Idan tsayin bututun kwarara ko bututu iri ɗaya ne, to ana canza lissafin Bernoulli zuwa:
