Ngā Rauemi Ture a Ākirīmete
Kua kite koe i tētahi kaipuke? Ki te mea kāore anō kia kite koe i tētahi, ko te tumanako kua kite koe i tētahi i runga i te pouaka whakaata, i te ipurangi rānei. Kāore he kaipuke he nui rawa te taumaha e totohu, engari ka taea e te toka iti, māmā. He aha i pērā ai? He tino māmā te whakautu mēnā e mārama ana koe ki ngā ariā o te pōrewarewa me te mātāpono a Ākimete.
I roto i te oranga o ia rā, ka kitea e tātou he iti ake te taumaha o ngā mea e whakanohoia ana ki roto i te wai, pērā i te toka, i tō te wā kāore te mea i roto i te wai. Tērā pea ka uaua ki te hiki i te toka mai i te mata o te whenua, engari ka ngāwari te hiki i te toka kotahi mai i te raro o te puna kaukau. Nā te kaha o te pūmau tēnei. Ka puta te pūmau nā te rerekētanga o te pēhanga wai i ngā hohonutanga rerekē. Ka piki ake te pēhanga wai i te hohonutanga, ka hohonu ake te wai, ka nui ake te pēhanga wai. Ina whakanohoia he mea ki roto i te wai, ka rerekē te pēhanga i waenga i te wai i runga ake i te mea me te wai i raro iho i te mea. He nui ake te pēhanga o te wai kei raro o te mea i te wai kei runga ake i te mea.
Pupuhi
I roto i te ahua, e kitea ana tētahi mea e mānu ana i roto i te wai. He nui ake te pēhanga o te wai i raro i te mea i te wai i runga ake i te mea. Nā te mea he hohonu ake te wai i raro i te mea i te wai i runga ake i te mea (h2 > h1).
F 2 = te kaha i tukuna e te wai ki raro o te mea, F 1 = te kaha i tukuna e te wai ki runga o te mea, A = te horahanga o te mata o te mea
Ko te rerekētanga i waenga i te F 2 me te F 1 ko te kaha katoa e tukuna ana e te wai ki runga i te mea, e mōhiotia ana ko te kaha pōrewarewa. Ko te rahi o te kaha pōrewarewa ko:
F te pūmautanga = F 2 − F 1
F whakakorikori = (ρ gh 2 A ) − (ρ gh 1 A)
F mānu = ρ g A ( h 2 − h 1 )
F te pūmautanga = ρ g A h
F te pūmautanga = ρ g V
Ngā Mōhiohio:
ρ = te mātotoru wai
g = te whakaterenga nā te kaha ā-papa
V = te rōrahi o te mea i roto i te wai
kia taea ai te tuhi i te whārite e kī ana i te rahi o te kaha pōturi (F pōturi) i runga ake nei penei:
F mānu = ρ F g V → m = ρ V
F te pūmau = m F g
F te pūmau = w F
m F g = w F = taumaha o te wai he rite te rōrahi ki te rōrahi o te mea kua totohu ki raro.
Te Mātāpono a Ākirīmete
I runga i te whārite i runga ake nei, ka taea e tātou te kī he ōrite te kaha pōrewarewa ki te taumaha o te wai kua nekehia, he ōrite te rōrahi o te wai kua nekehia ki te rōrahi o te mea kua rumakina ki roto i te wai.
Ki te rumakina tētahi mea ki roto i tētahi wai, ā, ka mānu, me te wāhanga anake o te mea e rukuhia ana, ko te rōrahi o te wai kua nekehia he rite ki te rōrahi o te wāhanga o te mea e rukuhia ana ki roto i te wai. Ahakoa te āhua, te rahi rānei o te mea, ka rite tonu te mea ka pā ki te katoa. Koinei te mahi a Ākirīmete (287-212 BC), e mōhiotia ana i nāianei ko te Mātāpono a Ākirīmete.
E mea ana te mātāpono a Ākimedes:
Ina rumakina katoatia, ka rumakina rānei tetahi mea ki roto i te wai, ka puta he kaha whakarunga (kaha pōrewarewa) ki runga i te mea, ā, ko te rahi o te kaha whakarunga (kaha pōrewarewa) he ōrite ki te taumaha o te wai kua nekehia.
Te Kōrero mō Ākimete
I noho a Ākirīmete i waenganui i te tau 287 me te tau 212 BC, ā, i tonoa ia e Kingi Hieron II ki te tirotiro mēnā he koura parakore te karauna i hangaia mō te kīngi. I te tīmatanga, i pōhēhē a Ākirīmete mēnā he koura parakore te karauna, mēnā rānei he konganuku kē atu kei roto. Ko te raruraru, he āhua koretake te karauna, ā, kāore e taea te whakangaro hei whakatau mēnā he koura parakore.
Ko te whakaaro mō te whakatau mēnā he koura parakore te karauna o te kīngi, ko te whakatau tuatahi i te taumaha motuhake o te karauna, kātahi ka whakataurite ki te taumaha motuhake o te koura. Mēnā he koura parakore te karauna, ko te taumaha motuhake o te karauna = te taumaha motuhake o te koura.
Ko te taumaha motuhake o tētahi mea ko te ōwehenga o te taumaha o te mea i te rangi ki te taumaha o te wai he rite te rōrahi ki te mea. E ai ki te pāngarau:
Me pēhea te whakatau i te taumaha o te wai he rite te rōrahi ki te rōrahi o tētahi mea?
E ai ki a Ākimete, ko te taumaha o te wai he rite te rōrahi ki te rōrahi o te mea = te rahi o te kaha pōrewarewa ina totohu te mea (kua rukuhia te mea katoa ki te wai). He rite tonu tēnei ki te taumaha o te mea i ngaro i te paunatia ki te wai. Nō reira:
Hei whakatau i te taumaha motuhake o te karauna, ka paunatia tuatahitia te karauna ki te hau (Taumaha o te Karauna i te Hau). Muri iho, ka makahia te karauna ki te wai, kātahi ka paunatia anō kia whiwhi ai i te taumaha o te karauna kua ngaro. Nō reira:
Kia whiwhihia te taumaha motuhake o te karauna, ka whakaritea ki te taumaha motuhake o te koura. Ko te taumaha motuhake o te koura he 19,3. Mena he rite te taumaha motuhake o te karauna ki te taumaha motuhake o te koura, ka hangaia te karauna ki te koura parakore. Heoi, ki te kore te karauna i hangaia ki te koura parakore, kāore te taumaha motuhake o te karauna e rite ki te taumaha motuhake o te koura.
He aha ngā kaipuke kāore e totohu?
Mena he iti iho te mātotoru o tētahi mea i te mātotoru o te wai, ka mānu te mea. I tetahi atu taha, ki te nui ake te mātotoru o tētahi mea i te mātotoru o te wai, ka totohu te mea. He mea hanga te nuinga o ngā kaipuke ki te rino me te maitai. Ko te mātotoru o te rino me te maitai = 7,8 x 103 kg /m3 , ko te mātotoru o te wai = 1,00 x 103 kg /m3 . Te āhua nei he nui ake te mātotoru o te rino me te maitai i te mātotoru o te wai. I tēnei wā, ko te taumaha motuhake o te rino me te maitai = 7,8. Me totohu ngā kaipuke i hangaia ki te rino me te maitai. He aha te take kāore ai e totohu ngā kaipuke? He iti ake te mātotoru katoa o te kaipuke i te mātotoru o te wai, o te wai moana rānei.
Ngā tauira pātai mō te ture a Archimedes
Tauira pātai 1:
Kei raro i te puna kaukau tētahi toka 40 kg. Mēnā ko te rōrahi o te toka = 0,2 m 3, he aha te kaha iti rawa e hiahiatia ana hei hiki i te toka?
Kōrero
E mōhiotia ana:
Taumaha o te kōhatu (m) = 40 Kg
Te rōrahi o te kōhatu (V) = 0,02 m3
Te mātotoru o te wai = 1000 Kg/ m3
Te whakaterenga nā te kaha ā-papa (g) = 10 m/s 2
I pātaihia: Te iti rawa o te F
Whakautu:
F te pūmau = w F
F te pūmau = m F g → m = pV
F te pūmautanga = ρ F g V
Te pūmautanga o te wai F = (1000 kg/m 3 )(10 m/s 2 )(0,02 m 3 )
F te pūmautanga = 200 kg m/s 2
F te pūmau = 200 N
Taumaha o te kōhatu (w) = mg
Taumaha o te kōhatu = (40 kg)(10 m/s 2 )
Taumaha o te kōhatu = 400 kg m/s 2
Taumaha o te kōhatu = 400 N
Te kaha iti rawa e hiahiatia ana hei hiki i te toka:
Taumaha o te kōhatu – Te kaha pōrewarewa = 400 N – 200 N = 200 Newton.
Tauira pātai 2:
Ko te taumaha o tētahi mea i te rangi = 5000 kg m/s² me te taumaha o tētahi mea i te wai = 4000 kg m/s² . Mēnā ko te mātotoru o te mea = 2000 kg/m³ , he aha te papatipu me te rōrahi o te mea? g = 10 m/ s²
Kōrero
Te whakaterenga nā te kaha ā-papa (g) = 10 m/s 2
Te mātotoru o te mea = 2000 kg/ m3
Te mātotoru o te wai = 1000 kg/ m3
Taumaha o te mea i te hau = 5000 kg m/s 2
Taumaha o te mea i roto i te wai = 4000 kg m/s 2
Te kaha pōrewarewa (F pōrewarewa) = Te taumaha o te mea i te rangi – Te taumaha o te mea i te wai
Te pūmautanga o te wai F = 5000 kg m/s 2 – 4000 kg m/s 2
F te pūmautanga = 1000 kg m/s 2
F te pūmau = Taumaha o te wai kua nekehia
F te pūmautanga = (papatipu o te wai)(g)
F te pūmau = (te rōrahi wai kua nekehia)(te mātotoru o te wai)(g)
Te rōrahi o te wai i nekehia = Te rōrahi o te mea i rukuhia ki te wai 😉
Nō reira, ko te rōrahi o te mea = 0,1 m 3
Te papatipu o te mea = ?
ρ = m / V
m = ρ V
m = ( 2000 kg / m 3 )( 0,1 m 3 )
m = 200 kg
Papatipu o te mea = 200 kg
Tauira pātai 3:
E hia te rahinga o te hēriuma e hiahiatia ana mēnā me hiki e te poihau te kawenga o te 500 kg?
Kōrero
Te mātotoru o te hauhā = 0,1786 kg/ m3
Te mātotoru o te hau = 1,293 kg/ m3
Te kaha pōrewarewa = Te taumaha o te hau kua nekehia = Te taumaha o te kawenga + Te taumaha o te hēriuma
Te kaha pōrewarewa = Taumaha o te kawenga + taumaha o te hēriuma
Te kaha pōrewarewa = (papatipu o te kawenga)(g) + (papatipu o te hēriuma)(g)
Te kaha pōrewarewa = (papatipu o te kawenga + papatipu o te heriuma)g —- whārite 1
Te kaha pūmau = Te taumaha o te hau kua nekehia
Te kaha pūmau = (papatipu o te hau kua nekehia)(g) —- whārite 2
Ka whakakotahi mātou i te whārite 1 me te whārite 2:
(papatipu o te kawenga + papatipu o te hēriuma)(g) = (papatipu o te hau kua nekehia)(g)
te papatipu o te kawenga + te papatipu o te hēriuma = te papatipu o te hau kua nekehia
500 kg + (ρ heriuma)(V heriuma) = (ρ hau)(V hau)
500 kg = (ρ hau)(V hau) – (ρ heriuma)(V heriuma)
Te rōrahi o te hau i nekehia (V hau) = Te rōrahi o te hēriuma e whakakī ana i te poihau (V hēriuma)
500 kg = (ρ hau – ρ helium)(V)
Koinei te rahinga iti rawa o te hēriuma e hiahiatia ana hei hiki i te taumaha i te mata o te Ao. Kia taea ai e te poihau te rere ki runga ake, me whakanui ake te rahinga o te hēriuma. Nā te mea ka heke te mātotoru o te hau i te teitei.
