Mafomula a Harmonic Vibration ndi Zitsanzo
Kugwedezeka kwa harmonic ndi nkhani yofunika kwambiri mu fizikisi, yomwe imawonekera kawirikawiri m'makambirano okhudza kuyenda kwa nthawi ndi nthawi, mafunde, ndi ntchito zauinjiniya. Titha kuipeza mu kuyenda kwa masika, ma pendulum (a ngodya zazing'ono), komanso kugwedezeka kwa mamolekyulu mu chinthu. Imatchedwa "harmonic" chifukwa kuyenda kwake kumatsatira dongosolo lokhazikika ndipo kumatha kupangidwa ndi ntchito za sine kapena cosine. Nkhaniyi ikufotokoza tanthauzo lalifupi, njira zazikulu, ndi mavuto a zitsanzo ndi mayankho kuti atithandize kumvetsetsa lingaliro la kugwedezeka kwa harmonic.
1. Kumvetsetsa Kugwedezeka kwa Harmonic
Kugwedezeka kosavuta kwa harmonic (SHM) ndi kayendedwe ka chinthu chodutsa malo ofanana ndi mphamvu yobwezeretsa yomwe kukula kwake kuli kofanana ndi kusamuka kwake ndipo nthawi zonse imayang'ana malo ofanana. Mwa masamu, makhalidwe ake akhoza kulembedwa motere:
F = −kx
Chizindikiro cha minus chimasonyeza kuti mphamvuyo ikupita moyang'anizana ndi njira yopatuka. Ngati chinthucho chakokedwa kumanja (positive x), mphamvu yobwezeretsayo ikupita kumanzere (negative), ndipo mosemphanitsa.
Machitidwe awiri omwe nthawi zambiri amatchulidwa ngati zitsanzo za GHS ndi awa:
1. Dongosolo la masika–kulemera (kulemera kumapeto kwa masika).
2. Pendulum yosavuta (ya ngodya zazing'ono, nthawi zambiri < 15°).
2. Kuchuluka Kofunika mu Harmonic Vibrations
Kuchuluka kwina komwe kumawonekera nthawi zonse mu GHS:
– Kupatuka (x): mtunda wa chinthucho kuchokera pamalo ofanana (m).
– Kuchuluka kwa mphamvu (A): kupotoka kwakukulu (m).
– Nthawi (T): nthawi yofunikira kuti munthu agwedezeke kwathunthu.
– Mafupipafupi (f): chiwerengero cha kugwedezeka pa sekondi (Hz).
– Kuthamanga kwa angular / mafupipafupi a angular (ω): gawo lofunikira mu sine/cosine equation (rad/s).
– Gawo (φ): limatsimikiza momwe kayendedwe kamayambira.
Ubale woyambira pakati pa nthawi ndi kuchuluka kwa nthawi:
f = 1/T
Ndipo ubale wa ω ndi T ndi f:
ω = 2πf = 2π/T
3. Harmonic Vibration Equation
Mawonekedwe onse a kupatuka monga momwe nthawi imagwiritsidwira ntchito angalembedwe:
x(t) = A tchimo(ωt + φ)
kapena
x(t) = A cos(ωt + φ)
Kusankha kwa sin/cos ndikolondola mofanana, kutengera mikhalidwe yoyambirira (monga pamene t = 0, malo ali mu amplitude kapena pa equilibrium point).
Liwiro ndi Kuthamanga
Velocity ndiye chiyambi choyamba cha kusamuka:
v(t) = dx/dt = Aω cos(ωt + φ) (ngati x = A sin)
kapena ingakhalenso ndi mtengo woipa malinga ndi mawonekedwe oyamba.
Kuthamanga ndi chochokera chachiwiri:
a(t) = d²x/dt² = −Aω² sin(ωt + φ)
Popeza x(t) = A sin(ωt + φ), ndiye:
a(t) = −ω² x(t)
Ichi ndi chinthu chofunikira kwambiri pa GHS: kuthamanga kwa liwiro kumakhala kofanana ndi kusuntha ndipo kumasiyana ndi komwe kukupita.
Liwiro Lalikulu ndi Kuthamanga Kwambiri
– v_max = Aω
– a_max = Aω²
Liwiro lalikulu kwambiri limachitika pamene chinthucho chadutsa mfundo yofanana (x = 0). Liwiro lalikulu kwambiri limachitika pa amplitude (x = ±A).
4. Nthawi ya Kugwedezeka mu Springs ndi Pendulums
A. Masika–Misa
Kuti mupeze kasupe wabwino kwambiri wokhala ndi nthawi yokhazikika ya kasupe k ndi kulemera m:
T = 2π √(m/k)
ω = √(k/m)
Izi zikutanthauza kuti kulemera kukakhala kwakukulu, nthawi imakhala yayikulu (yocheperako). K (kasupe wolimba), nthawi imakhala yochepa (yofulumira).
B. Pendulum Yosavuta (Ngodya Yaing'ono)
Kwa pendulum yokhala ndi utali wa chingwe L ndi mphamvu yokoka g:
T = 2π √(L/g)
ω = √(g/L)
Chochititsa chidwi n'chakuti, poyang'ana mbali yaying'ono, nthawiyo siidalira kulemera kwa pendulum, koma kutalika ndi mphamvu yokoka.
5. Mphamvu mu Harmonic Vibrations
Mu GHS, mphamvu yonse imakhalabe yofanana (ngati palibe kukangana):
– Mphamvu yomwe ingakhalepo ya masika:
Ep = ½ kx²
- Mphamvu ya Kinetic:
Ek = ½ mv²
– Mphamvu yonse ya makina:
E = Ek + Ep = ½ kA²
Pamene x = ±A, v = 0 kotero mphamvu ndi yamphamvu kwathunthu. Pamene x = 0, Ep = 0 ndipo mphamvu ndi ya kinetic kwathunthu.
6. Mafunso ndi Kukambirana Zitsanzo
Chitsanzo 1: Kudziwa Nthawi ya Masika
Kulemera kwa 0,5 kg kumapachikidwa pa kasupe wokhala ndi k yokhazikika = 200 N/m. Dziwani nthawi ya kugwedezeka!
Kuperekedwa: m = 0,5 kg, k = 200 N/m
Anafunsa: T
Yankho:
T = 2π √(m/k)
= 2π √(0,5 / 200)
= 2π √(0,0025)
= 2π (0,05)
= 0,1π s
≈ 0,314 s
Kotero, nthawi ya kugwedezeka kwa masika ndi pafupifupi 0,314 s.
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Chitsanzo 2: Kudziwa Mafupipafupi ndi ω
Kuchokera pa funso 1, dziwani kuchuluka kwa (f) ndi ω!
Yankho:
f = 1/T = 1/0,314 ≈ 3,18 Hz
ω = 2π/T = 2π/0,314 ≈ 20 rad/s
(kapena mwachindunji ω = √(k/m) = √(200/0,5) = √400 = 20 rad/s)
Kotero, ma frequency ndi pafupifupi 3,18 Hz ndipo ω = 20 rad/s.
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Chitsanzo 3: Equation yopotoka
Chinthu chimagwedezeka mofanana ndi kutalika kwa 0,10 m ndi ω = 5 rad/s. Pa t = 0, chinthucho chili pamlingo woyenera ndipo chikuyenda bwino. Dziwani equation yosuntha!
Kuperekedwa: A = 0,10 m, ω = 5 rad/s
Mikhalidwe yoyambirira: t = 0 → x = 0 ndipo v yoyambirira ndi yabwino.
Ngati x(0) = 0, mawonekedwe oyenera ndi sine:
x(t) = A tchimo(ωt)
Popeza v(t) = Aω cos(ωt), ndiye v(0) = Aω cos(0) = Aω yabwino, moyenerera.
Kotero:
x(t) = 0,10 sin(5t) (mita)
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Chitsanzo 4: Liwiro Lalikulu ndi Kuthamanga Kwambiri
Kuchokera ku chitsanzo chachitatu, dziwani v_max ndi a_max.
Yankho:
v_max = Aω = 0,10 × 5 = 0,50 m/s
a_max = Aω² = 0,10 × 25 = 2,5 m/s²
Kotero, v_max = 0,50 m/s ndipo a_max = 2,5 m/s².
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Chitsanzo 5: Mphamvu mu Kasupe
Kasupe wokhala ndi k = 100 N/m amanjenjemera ndi amplitude ya A = 0,20 m. Werengani mphamvu zake zonse zamakina!
Yankho:
E = ½ kA²
= ½ (100)(0,20)²
= 50 × 0,04
= 2 J
Mphamvu yonse ya makina ya kugwedezeka ndi ma joules awiri.
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Chitsanzo 6: Nthawi ya Pendulum
Pendulum yosavuta ndi 1 m kutalika. Ngati g = 10 m/s², dziwani nthawi yake.
Yankho:
T = 2π √(L/g)
= 2π √(1/10)
= 2π √0,1
≈ 2π (0,316)
≈ 1,99 s
Kotero, nthawi ya pendulum ndi pafupifupi 2,0 s.
7. Mapeto
Kusinthasintha kosavuta kwa harmonic ndi kayendedwe kofunikira kwambiri mu fizikisi. Chinsinsi chomvetsetsa izi chili mu ubale womwe ulipo pakati pa mphamvu yobwezeretsa ndi kusamuka (F = −kx), ndi chitsanzo chake cha masamu, chomwe chimatsatira ntchito ya sine/cosine. Mukamvetsetsa njira za nthawi ya masika ndi pendulum, ma equation a displacement–velocity–acceleration, ndi lingaliro la mphamvu, mudzapeza kuti n'zosavuta kuthetsa mavuto okhudzana ndi kugwedezeka ndi mafunde.
Ngati mukufuna, nditha kuwonjezera mafunso ena oyeserera (popanda kukambirana kaye) kapena kupanga chidule cha njira zokonzekera mayeso.
