Ngā Tātai me ngā Tauira o te Wiri Ā-Haramai
He kaupapa matua te wiri ā-pūtaiao, e puta pinepine ana i roto i ngā kōrero mō te nekehanga ā-wā, ngā ngaru, me ngā tono hangarau. Ka kitea i roto i te nekehanga o ngā puna, ngā pendulum (mō ngā koki iti), tae noa ki te wiri o ngā ngota i roto i te mea. Ka kiia ko te "ā-pūtaiao" nā te mea e whai ana tōna nekehanga i tētahi tauira auau, ā, ka taea te whakatauira me ngā mahi sine, cosine rānei. Ka matapakihia e tēnei tuhinga tētahi whakamāramatanga poto, ngā tātai matua, me ngā tauira rapanga me ngā otinga hei āwhina i a tātou ki te mārama ki te ariā o te wiri ā-pūtaiao.
1. Te Mārama ki ngā Wiri Harmonic
Ko te wiri ā-whakaaro māmā (SHM) te nekehanga whakamuri me te whakamuri o tētahi mea i tua atu i tētahi pūwāhi taurite me te kaha whakaora e rite ana te rahi ki te nekehanga, ā, ko tōna ahunga e anga tonu ana ki te pūwāhi taurite. Mā te pāngarau, ka taea te tuhi i ōna āhuatanga penei:
F = −kx
Ko te tohu tango e tohu ana kei te ritenga kē te ahunga o te kaha ki te ahunga peka. Ki te tōia te mea ki te taha matau (x pai), ka tōia te kaha whakaora ki te taha maui (kino), ā, ko te ritenga kē.
Ko ngā pūnaha e rua e whakahuatia ana hei tauira o te GHS ko:
1. Pūnaha puna-papatipu (papatipu i te mutunga o te puna).
2. Pendulum māmā (mō ngā koki iti, te nuinga o te wā < 15°). 2. Ngā Rahinga Hira i roto i te Wiri Āhuahira Ko ētahi rahinga e puta tonu ana i roto i te GHS: - Te rerekētanga (x): te tawhiti o te mea mai i te pūwāhi taurite (m). - Te whānui (A): te rerekētanga mōrahi (m). - Te wā (T): te wā e hiahiatia ana mō te kotahi wiri katoa (s). - Te auau (f): te maha o ngā wiri ia hekona (Hz). - Te tere koki / te auau koki (ω): he tawhā nui i roto i te whārite sine/cosine (rad/s). - Te wāhanga (φ): te whakatau i ngā āhuatanga tīmatanga o te nekehanga. Ko te whanaungatanga taketake i waenga i te wā me te auau: f = 1/T Me te whanaungatanga o ω ki a T me f: ω = 2πf = 2π/T 3. Whārite Wiri Āhuahira Ka taea te tuhi i te āhua whānui o te rerekētanga hei mahi a te wā: x(t) = A sin(ωt + φ) x(t) rānei = A cos(ωt + φ)
He tika anō te kōwhiringa o te sin/cos, i runga i ngā āhuatanga tīmatanga (hei tauira, i te t = 0, kei te kaha te tūnga, kei te pūwāhi taurite rānei). Te Tere me te Whakaterenga Ko te tere te pānga tuatahi o te nekehanga: v(t) = dx/dt = Aω cos(ωt + φ) (mēnā he A sin) ka taea rānei te kino i runga i te āhua tīmatanga. Ko te whakaterenga te pānga tuarua: a(t) = d²x/dt² = −Aω² sin(ωt + φ) Nā te mea he A sin(ωt + φ), kāti: a(t) = −ω² x(t) He āhuatanga matua tēnei o te GHS: he ōrite te whakaterenga ki te nekehanga, ā, he ritenga kē te ahunga. Te Tere Mōrahi me te Whakaterenga Mōrahi - v_max = Aω - a_max = Aω² Ka puta te tere mōrahi ina whiti te mea i te pūwāhi taurite (x = 0). Ka puta te whakaterenga mōrahi ina kei roto i te kaha (x = ±A). 4. Wā o te Wiri i roto i ngā Puna me ngā Pendulum A. Puna–Papatipu Mō tētahi puna pai me te pūmau puna k me te papatipu m: T = 2π √(m/k) ω = √(k/m) Ko te tikanga tēnei, ko te nui ake o te papatipu, ko te nui ake o te wā (pōturi ake). Ko te nui ake o te k (ko te pakari ake o te puna), ko te iti ake o te wā (tere ake). B. Pendulum Māmā (Koki Iti) Mō tētahi pendulum me te roa o te aho L me te whakaterenga ā-papatipu g: T = 2π √(L/g) ω = √(g/L) He mea whakamīharo, i roto i te whakatata koki iti, kāore te wā e whakawhirinaki ana ki te papatipu o te pendulum, engari e whakawhirinaki ana ki te roa me te kaha ā-papatipu. 5. Pūngao i roto i te Wiri Ā-Hangarau I roto i te GHS, ka noho pūmau te pūngao katoa (mēnā kāore he waku): - Pūngao pūmanawa o te puna: Ep = ½ kx² - Pūngao nekeneke: Ek = ½ mv² - Pūngao miihini katoa: E = Ek + Ep = ½ kA² Ina x = ±A, v = 0, nō reira he pūmanawa katoa te pūngao. Ina x = 0, Ep = 0, ā, he nekeneke katoa te pūngao. 6. Tauira Pātai me te Kōrero Tauira 1: Te Whakatau i te Wā Puna He papatipu 0,5 kg te taumaha e iri ana ki runga i te puna me te pūmau k = 200 N/m. Whakatauhia te wā wiri! Hoatu: m = 0,5 kg, k = 200 N/m I pātaihia: T Whakautu: T = 2π √(m/k) = 2π √(0,5 / 200) = 2π √(0,0025) = 2π (0,05) = 0,1π s ≈ 0,314 s
Nō reira, ko te wā o te wiri o te puna he tata ki te 0,314 s. --- Tauira 2: Te Whakatau i te Auautanga me te ω Mai i te pātai 1, whakatauhia te auautanga (f) me te ω! Whakautu: f = 1/T = 1/0,314 ≈ 3,18 Hz ω = 2π/T = 2π/0,314 ≈ 20 rad/s (he tika rānei ω = √(k/m) = √(200/0,5) = √400 = 20 rad/s) Nō reira, ko te auautanga he tata ki te 3,18 Hz me te ω = 20 rad/s. --- Tauira 3: Whārite Whakarerekētanga Ka wiri tetahi mea i te āhua harmonic me te kaha o te 0,10 m me te ω = 5 rad/s. I te t = 0, kei te pūwāhi taurite te mea, ā, ka neke i te ahunga pai. Whakatauhia te whārite whakarerekētanga! Hoatu: A = 0,10 m, ω = 5 rad/s Ngā āhuatanga tīmatanga: t = 0 → x = 0, ā, he pai te v tīmatanga. Mena ko x(0) = 0, ko te āhua tika ko te sine: x(t) = A sin(ωt) Nā te mea ko v(t) = Aω cos(ωt), ko v(0) = Aω cos(0) = Aω he pai. Nō reira: x(t) = 0,10 sin(5t) (mita) --- Tauira 4: Te Tere Mōrahi me te Whakaterenga Mōrahi Mai i te tauira 3, whakatauhia ko v_max me a_max. Whakautu: v_max = Aω = 0,10 × 5 = 0,50 m/s a_max = Aω² = 0,10 × 25 = 2,5 m/s² Nō reira, ko v_max = 0,50 m/s me a_max = 2,5 m/s². --- Tauira 5: Pūngao i roto i te Puna Ka wiri tetahi puna me te k = 100 N/m me te kaha o A = 0,20 m. Tātaihia tōna pūngao miihini katoa! Whakautu: E = ½ kA² = ½ (100)(0,20)² = 50 × 0,04 = 2 J Ko te pūngao miihini katoa o te wiri he 2 joules. --- Tauira 6: Wā Pendulum He 1 m te roa o te pendulum māmā. Mena ko g = 10 m/s², tātaihia tōna wā. Whakautu: T = 2π √(L/g) = 2π √(1/10) = 2π √0,1 ≈ 2π (0,316) ≈ 1,99 s Nō reira, ko te wā o te pendulum he tata ki te 2,0 s. 7. Whakamutunga He nekehanga ā-wā tino taketake te wiri harmonic māmā i roto i te ahupūngao. Ko te mea nui ki te mārama ki tēnei mea kei roto i te whanaungatanga i waenga i te kaha whakaora e rite ana ki te nekehanga (F = −kx), me tana tauira pāngarau e whai ana i te mahi sine/cosine. Mā te mārama ki te tātai wā mō ngā puna me ngā pendulum, ngā whārite nekehanga-tere-whakatere, me te ariā o te pūngao, ka māmā ake māu te mahi i ngā rapanga e pā ana ki ngā wiri me ngā ngaru.
Ki te hiahia koe, ka taea e au te tāpiri i ētahi atu pātai whakaharatau (me te kore kōrero i te tuatahi) ka taea rānei e au te waihanga i tētahi putanga whakarāpopoto o ngā tātai kua rite mō te whakamātautau.
