He tuhinga mō te mātāpono o Archimedes
Kāore he kaipuke nui e totohu, engari ka totohu te kōhatu iti. He aha ai? He māmā noa te whakautu mēnā e mārama ana koe ki te 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 whakaurua ana ki roto i te wai, pērā i te toka, i ērā atu mea kāore i roto i te wai. Tērā pea he uaua ki te hiki ake i te kōhatu mai i te whenua, engari ka ngāwari te hiki ake i te kōhatu kotahi mai i te raro o te wai moana. Nā te kaha pōkaikaha tēnei. Ka puta te pōkaikaha nā te rerekētanga o te pēhanga wai i ngā hohonutanga rerekē. Ka piki ake te pēhanga wai i te hohonutanga, ka matotoru ake te wai, ka nui ake te pēhanga o te wai. Ina whakaurua he mea ki roto i te wai, ka rerekē te pēhanga i waenga i te wai kei runga o te mea me te wai kei raro o te mea. He nui ake te pēhanga o te wai kei raro o te mea i te wai kei runga o te mea.
I roto i te pikitia, ka kite koe i 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 kei runga. Nā te mea he nui ake te hohonutanga o te wai i raro i te mea i te wai i runga ake i te mea (p2 > h1).
Ko te nui o te pēhanga wai i te hohonutanga o te h 2 ko:
Ko te nui o te pēhanga wai i te hohonutanga o te h 1 ko:
F 2 = te kaha e pā ana ki te wai i te raro o te mea, F 1 = te kaha e pā ana ki te wai i te tihi 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 pōturi. Ko te nui o te pōturi ko:
F te pūmautanga = F 2 − F 1
F te whakaea = (ρ gh 2 A) − (ρ gh 1 A)
F te whakaea = ρ g A (h 2 − h 1 )
F te pūmautanga = ρ F g A h
F te pūmautanga = ρ F g V
ρ F = kiato wai, g = whakaterenga ā-papa, V = rōrahi o ngā mea i roto i te wai
Nō reira, ka taea e tātou te tuhi i te whārite e whakaatu ana i te nui o te pūmau (F pūmau) i runga ake nei:
F te kaha = ρ F g V → m = ρ V
F te pūmau = m F g
F te pūmau = w F
m F g = w F = te taumaha o te wai e rite ana te rōrahi ki te rōrahi o te mea kua toua.
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 e nekehia ana, he rite te rōrahi o te wai e nekehia ana ki te rōrahi o te mea kua toua ki roto i te wai.
Ki te whakaurua te mea ki roto i te wai, e mānu ana, ā, he wāhanga noa iho te wāhanga o te mea e whakaurua ana,
kātahi ko te rōrahi o te wai e nekehia ana = te rōrahi o te wāhanga o te mea kua tōia ki roto i te wai. Ahakoa he aha te mea me tōna āhua, ka pāngia te katoa e te mea kotahi. Koinei te hua o te mahi a Ākimedes (287-212 BC), e mōhiotia ana ko te Mātāpono a Ākimedes.
E mea ana te mātāpono a Archimedes:
Ina tōia katoatia, ka tōia wāhanga rānei tētahi mea ki roto i tētahi wai, ka tukuna e te wai he kaha whakarunga (kaha pōrewarewa) ki te mea, ā, ko te nui o te kaha whakarunga (kaha pōrewarewa) he ōrite ki te taumaha o te wai e nekehia ana.
Te kōrero mō Ākimete
I noho a Ākirīmete i waenganui i te tau 287-212 BC, ā, i tonoa ia e Kingi Hieron II ki te tirotiro mēnā he koura parakore te karauna i hangaia mō te Kīngi. Hei kimi mēnā he koura parakore te karauna, he konganuku kē atu rānei kei roto i te karauna, i pōhēhē a Ākirīmete i te tuatahi. Ko te raruraru, he āhua koretake te āhua o te karauna, ā, kāore e taea te whakangaro tuatahi kia taea ai te whakatau mēnā he koura parakore te karauna.
Ko te whakaaro mō te whakatau mēnā he koura parakore te karauna, ko te whakatau tuatahi i te taumaha 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 i waenga i te taumaha o te mea i te rangi me te taumaha o te wai he rite te rōrahi ki te rōrahi o ngā mea. I tuhia i roto i te pāngarau:
Me pēhea te whakatau i te taumaha o te wai he rite te rōrahi ki te rōrahi o ngā mea?
E ai ki a Ākirīmete, ko te taumaha o te wai he rite te rōrahi ki te rōrahi o tētahi mea = te nui o te kaha pōrewarewa ina totohu te mea (ka tōia te wāhanga katoa o te mea ki te wai). He rite tonu tēnei ki te taumaha o ngā mea i ngaro ina paunatia ki te wai. Nō reira:
Hei whakatau i te taumaha motuhake o te karauna, ka paunatia tuatahitia te karauna i te rangi (te taumaha o te karauna i te rangi). Kātahi ka whakaurua te karauna ki te wai, kātahi ka paunatia anō kia whiwhi ai i te Taumaha i Ngaro o te Karauna. Nō reira:
Kia whiwhihia te taumaha motuhake o te karauna, ka whakaritea ki te taumaha motuhake o te koura. Te taumaha motuhake o te koura = 19.3. Mena ko te taumaha motuhake o te karauna = te taumaha motuhake o te koura, ka hangaia te karauna ki te koura parakore. Engari 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 te 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 te kaipuke. He aha te take kāore ai te kaipuke e totohu? He iti ake te mātotoru katoa o te kaipuke i te mātotoru o te wai, o te wai moana rānei.
Tauira raruraru 1:
Kei te raro o te puna tētahi kōhatu he 40 kg te taumaha. Mēnā ko te rōrahi o te kōhatu = 0.2 m3, he aha te kaha iti rawa e hiahiatia ana hei hiki i te kōhatu?
Mōhiotia:
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 o te kaha ā-papa (g) = 10 m/s 2
E hiahiatia ana: F iti rawa
Rongoā:
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 )
Te pūmautanga o te F = 200 kg m/s 2
F te pūmau = 200 N
taumaha o te kōhatu (w) = mg
te 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 kōhatu:
taumaha o te kōhatu – te kaha pōrewarewa = 400 N – 200 N = 200 N
Tauira raruraru 2:
Ko te taumaha o te mea i roto i te wai = 5000 kg m/s² me te taumaha o te mea i roto i te wai = 4000 kg m/s² . Mena ko te mātotoru o te mea = 2000 kg/m³ , he aha te papatipu me te rōrahi o te mea? g = 10 m/ s²
otinga
Te whakaterenga o 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 = te 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 kua nekehia = te rōrahi o te mea i roto i te wai
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
Taumaha o te mea = 200 kg
Tauira raruraru 3:
E hia te rahinga o te hēriuma e hiahiatia ana mēnā me hiki e te poihau te 500 kg o te kawenga?
Rongoā:
Te mātotoru o te hēriuma = 0.1786 kg/ m3
Te mātotoru o te hau = 1.293 kg/ m3
Te kaha pūmau = te taumaha o te hau kua nekehia = te taumaha o te mea + te taumaha o te hēriuma
Te kaha pūmau = te taumaha o te mea + te taumaha o te hēriuma
Te kaha pōrewarewa = (papatipu o te kawenga)(g) + (papatipu o te herium)(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 i 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 i 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 wai)(V wai)
500 kg = (ρ wai)(V wai) – (ρ hēriuma)(V hēriuma)
Te rōrahi o te hau i nekehia (V hau) = Te rōrahi o te hēriuma i roto i te poihau (V hēriuma)
500 kg = (ρ wai – ρ helium)(V)
Koinei te rahinga iti rawa o te hēriuma e hiahiatia ana hei hiki i ngā taumaha i runga i te mata o te whenua. Kia rere teitei ake ai te poihau, me tāpiri te rahinga o te hēriuma. Me whakanui ake te rahinga o te hēriuma nā te mea ka heke te mātotoru o te hau i te teitei.
