Nā Haʻilula Haʻalulu Harmonic a me nā Laʻana
He kumuhana koʻikoʻi ka haʻalulu harmonic i ka physics, e ʻike pinepine ʻia ana i nā kūkākūkā ʻana o ka neʻe ʻana o ka manawa, nā nalu, a me nā noi ʻenekinia. Hiki iā mākou ke loaʻa iā ia ma ka neʻe ʻana o nā pūnāwai, nā pendulums (no nā kihi liʻiliʻi), a me ka haʻalulu o nā molekala i loko o ka mea. Ua kapa ʻia ʻo "harmonic" no ka mea e hahai ana kona neʻe ʻana i kahi ʻano maʻamau a hiki ke hoʻohālike ʻia me nā hana sine a cosine paha. Kūkākūkā kēia ʻatikala i kahi wehewehe pōkole, nā ʻano hana koʻikoʻi, a me nā pilikia hoʻohālike me nā hoʻonā e kōkua iā mākou e hoʻomaopopo i ke kumumanaʻo o ka haʻalulu harmonic.
1. Ke Hoʻomaopopo ʻana i nā Haʻalulu Harmonic
ʻO ka haʻalulu harmonic maʻalahi (SHM) ka neʻe ʻana i mua a i mua o kahi mea ma mua o kahi kiko kaulike me kahi ikaika hoʻihoʻi nona ka nui e like me ka neʻe ʻana a nona ke kuhikuhi e kū mau ana i ke kiko kaulike. Ma ke ʻano makemakika, hiki ke kākau ʻia kona mau ʻano penei:
F = −kx
Hōʻike ka hōʻailona minus i ke kuhikuhi o ka ikaika e kūʻē i ke kuhikuhi o ka deviation. Inā huki ʻia ka mea i ka ʻākau (maikaʻi x), aia ka ikaika hoʻihoʻi i ka hema (maikaʻi ʻole), a ʻo ka hope.
ʻO nā ʻōnaehana ʻelua i ʻōlelo pinepine ʻia he mau hiʻohiʻona o GHS:
1. ʻŌnaehana puna-nuipaʻa (nuipaʻa ma ka hopena o ka puna).
2. Pendulum maʻalahi (no nā kihi liʻiliʻi, maʻamau < 15°). 2. Nā nui koʻikoʻi i ka Harmonic Vibration ʻO kekahi mau nui e ʻike mau ʻia ma GHS: - Deviation (x): ka mamao o ka mea mai ke kiko kaulike (m). - Amplitude (A): ka deviation kiʻekiʻe loa (m). - Period (T): ka manawa e pono ai no hoʻokahi vibration piha (s). - Frequency (f): ka helu o nā vibrations i kekona (Hz). - Angular velocity / angular frequency (ω): kahi parameter koʻikoʻi i ka sine/cosine equation (rad/s). - Phase (φ): hoʻoholo i nā kūlana mua o ka neʻe ʻana. ʻO ka pilina kumu ma waena o ka wā a me ke alapine: f = 1/T A ʻo ka pilina o ω me T a me f: ω = 2πf = 2π/T 3. Harmonic Vibration Equation Hiki ke kākau ʻia ke ʻano laulā o ka deviation ma ke ʻano he hana o ka manawa: x(t) = A sin(ωt + φ) a i ʻole x(t) = A cos(ωt + φ)
ʻO ke koho ʻana o sin/cos he pololei like ia, ma muli o nā kūlana mua (e.g., ma t = 0, aia ke kūlana ma ka amplitude a i ʻole ma ke kiko kaulike). Ka wikiwiki a me ka wikiwiki ʻO ka wikiwiki ka derivative mua o ka neʻe ʻana: v(t) = dx/dt = Aω cos(ωt + φ) (inā x = A sin) a i ʻole hiki ke maikaʻi ʻole e like me ke ʻano mua. ʻO ka wikiwiki ka derivative lua: a(t) = d²x/dt² = −Aω² sin(ωt + φ) ʻOiai ʻo x(t) = A sin(ωt + φ), a laila: a(t) = −ω² x(t) He waiwai koʻikoʻi kēia o GHS: kūlike ka wikiwiki me ka neʻe ʻana a kūʻē i ke kuhikuhi. Ka wikiwiki kiʻekiʻe a me ka wikiwiki kiʻekiʻe - v_max = Aω - a_max = Aω² Loaʻa ka wikiwiki kiʻekiʻe ke hala ka mea i ke kiko kaulike (x = 0). Loaʻa ka wikiwiki kiʻekiʻe ke hele i ka amplitude (x = ±A). 4. Ka Wā o ka Haʻalulu i nā Puna a me nā Pendulums A. Puna–Mass No kahi puna kūpono me ke kūpaʻa puna k a me ke kaumaha m: T = 2π √(m/k) ω = √(k/m) ʻO ke ʻano kēia, ʻo ka nui o ke kaumaha, ʻo ka nui o ke kau (lohi). ʻO ka nui o k (ʻoi aku ka paʻakikī o ka puna), ʻoi aku ka liʻiliʻi o ke kau (wikiwiki). B. Pendulum Maʻalahi (Huina Liʻiliʻi) No kahi pendulum me ka lōʻihi o ke kaula L a me ka wikiwiki o ka umekaumaha g: T = 2π √(L/g) ω = √(g/L) ʻO ka mea hoihoi, i ka hoʻokokoke ʻana o ke kihi liʻiliʻi, ʻaʻole hilinaʻi ka wā i ka nui o ka pendulum, hilinaʻi wale ia i ka lōʻihi a me ke umekaumaha. 5. Ikehu i ka Haʻalulu Harmonic Ma GHS, e mau ana ka ikehu holoʻokoʻa (inā ʻaʻohe friction): - Ikehu hiki o ka pūnāwai: Ep = ½ kx² - Ikehu kinetic: Ek = ½ mv² - Ikehu mechanical holoʻokoʻa: E = Ek + Ep = ½ kA² Ke x = ±A, v = 0 no laila he hiki āpau ka ikehu. Ke x = 0, Ep = 0 a he kinetic āpau ka ikehu. 6. Nā nīnau hoʻohālike a me ke kūkākūkā ʻana Laʻana 1: Ke hoʻoholo nei i ka wā puna Ua kau ʻia kahi nuipa o 0,5 kg ma luna o kahi pūnāwai me kahi k mau = 200 N/m. E hoʻoholo i ka wā haʻalulu! Hāʻawi ʻia: m = 0,5 kg, k = 200 N/m Nīnau ʻia: T Pane: T = 2π √(m/k) = 2π √(0,5 / 200) = 2π √(0,0025) = 2π (0,05) = 0,1π s ≈ 0,314 s
No laila, ʻo ka manawa o ka haʻalulu puna ma kahi o 0,314 s. --- Laʻana 2: Ke hoʻoholo nei i ka alapine (frequency) a me ω Mai ka nīnau 1, e hoʻoholo i ke alapine (f) a me ω! Pane: f = 1/T = 1/0,314 ≈ 3,18 Hz ω = 2π/T = 2π/0,314 ≈ 20 rad/s (a i ʻole pololei ω = √(k/m) = √(200/0,5) = √400 = 20 rad/s) No laila, ʻo ke alapine ma kahi o 3,18 Hz a ʻo ω = 20 rad/s. --- Laʻana 3: Hoʻohālikelike Deviation Hoʻoluliluli harmonic kahi mea me ka amplitude o 0,10 m a me ω = 5 rad/s. Ma t = 0, aia ka mea ma ke kiko kaulike a neʻe i ke kuhikuhi maikaʻi. E hoʻoholo i ka hoʻohālikelike deviation! Hāʻawi ʻia: A = 0,10 m, ω = 5 rad/s Nā kūlana mua: t = 0 → x = 0 a he maikaʻi ka v mua. Inā x(0) = 0, ʻo ke ʻano kūpono he sine: x(t) = A sin(ωt) ʻOiai ʻo v(t) = Aω cos(ωt), a laila v(0) = Aω cos(0) = Aω he maikaʻi, pēlā. Pēlā: x(t) = 0,10 sin(5t) (mika) --- Laʻana 4: Ka wikiwiki kiʻekiʻe loa a me ka wikiwiki kiʻekiʻe loa Mai ka laʻana 3, e hoʻoholo iā v_max a me a_max. Pane: v_max = Aω = 0,10 × 5 = 0,50 m/s a_max = Aω² = 0,10 × 25 = 2,5 m/s² No laila, v_max = 0,50 m/s a me a_max = 2,5 m/s². --- Laʻana 5: Ikehu i loko o kahi Puna He puna me k = 100 N/m e haʻalulu me ka amplitude o A = 0,20 m. E helu i kona ikehu mechanical holoʻokoʻa! Pane: E = ½ kA² = ½ (100)(0,20)² = 50 × 0,04 = 2 J ʻO ka ikehu mechanical holoʻokoʻa o ka haʻalulu he 2 joules. --- Laʻana 6: Ka Wā Pendulum He 1 m ka lōʻihi o kahi pendulum maʻalahi. Inā g = 10 m/s², e hoʻoholo i kona wā. Pane: T = 2π √(L/g) = 2π √(1/10) = 2π √0,1 ≈ 2π (0,316) ≈ 1,99 s No laila, ʻo ka wā o ka pendulum ma kahi o 2,0 s. 7. Hopena ʻO ka haʻalulu harmonic maʻalahi kahi neʻe periodic kumu loa i ka physics. ʻO ke kī i ka hoʻomaopopo ʻana iā ia aia i ka pilina ma waena o ka ikaika hoʻihoʻi e kūlike ana i ka neʻe ʻana (F = −kx), a me kāna kumu hoʻohālike makemakika e hahai ana i ka hana sine/cosine. Ma ka hoʻomaopopo ʻana i ke ʻano o ka wā no nā pūnāwai a me nā pendulums, nā hoʻohālikelike neʻe-wikiwiki-wikiwiki, a me ke kumumanaʻo o ka ikehu, e maʻalahi iā ʻoe ke hana ma nā pilikia e pili ana i nā haʻalulu a me nā nalu.
Inā makemake ʻoe, hiki iaʻu ke hoʻohui i nā nīnau hoʻomaʻamaʻa hou (me ka ʻole o ke kūkākūkā mua ʻana) a i ʻole e hana i kahi mana hōʻuluʻulu manaʻo o nā haʻilula i mākaukau no ka hoʻokolohua.
