Ifomula Ye-Angular Momentum
I-Pendahuluan
Umfutho we-angular ungumqondo obalulekile ku-physics ohlobene nokunyakaza kokujikeleza kwento. Lo mqondo ufana nomfutho oqondile ekunyakazeni kokuhumusha. Umfutho we-angular udlala indima ebalulekile emikhakheni ehlukahlukene ye-physics, kusukela kuma-mechanics akudala kuya kuma-quantum mechanics. Lesi sihloko sizoxoxa ngencazelo yomfutho we-angular, amafomula ahlobene, ukusetshenziswa ekuphileni kwansuku zonke, kanye nezibonelo zokujulisa ukuqonda.
Incazelo ye-Angular Momentum
I-Angular momentum iyinani le-vector elichaza ukuthambekela kwento ukuqhubeka ijikeleza iphuzu noma i-axis. I-Angular momentum (\(\vec{L}\)) incike ezintweni ezimbili eziyinhloko: i-linear momentum (\(\vec{p}\)) kanye nesikhundla esihlobene (\(\vec{r}\)) sephuzu lokubhekisela. I-Angular momentum ichazwa ngokuthi:
\[ \vec{L} = \vec{r} \times \vec{p} \]
Kuphi:
– \(\vec{L}\) yi-angular momentum.
– \(\vec{r}\) yivektha yesikhundla maqondana nephuzu lokubhekisela.
– \(\vec{p}\) yi-linear momentum (\(\vec{p} = m \vec{v}\), lapho \(m\) kuyisisindo kanye \(\vec{v}\) kuyijubane).
– \(\izikhathi\) imelela umkhiqizo ohlanganisa amavektha amabili.
Ifomula Ye-Angular Momentum
Kumzimba oqinile ojikeleza ngejubane le-angular (\(\omega\)) mayelana ne-axis eqondile, i-angular momentum (\(L\)) ingachazwa kanje:
\[ L = I \omega \]
Kuphi:
– \(L\) umfutho we-angular.
– \(I\) yisikhathi sokungapheleli kwento ezungeze i-axis yokujikeleza.
– \(\omega\) yijubane le-angular.
Isikhathi Sokungabi Naso
Umzuzu we-inertia (\(I\)) uyisilinganiso sokumelana kwento nezinguquko ekuhambeni kwayo kokujikeleza. Umzuzu we-inertia uncike ekusabalaleni kwesisindo sento maqondana ne-axis yokujikeleza. Entweni eqinile, umzuzu we-inertia ungabalwa kusetshenziswa ifomula:
\[ I = \sum m_i r_i^2 \]
Kuphi:
– \(m_i\) isisindo senhlayiya ye-\(i\)th.
– \(r_i\) ibanga le-particle \(i\)th ukusuka ku-axis yokujikeleza.
Ezintweni ezilula, umzuzu wokungahlali kahle unefomula yawo. Ezinye izibonelo yilezi:
– Isilinda Esingenalutho: \(I = mr^2\)
– Isilinda Esigcwele: \(I = \frac{1}{2} mr^2\)
– I-Full Sphere: \(I = \frac{2}{5} mr^2\)
Isimiso Sokulondolozwa Kwejubane Eliyindilinga
Isimiso sokulondolozwa komfutho we-angular sithi uma kungekho torque yangaphandle esebenza ohlelweni, umfutho we-angular ophelele wesistimu uzohlala ungaguquki. Lokhu kusho ukuthi:
\[ \vec{L}_{start} = \vec{L}_{end} \]
noma
\[ I_{initial} \omega_{initial} = I_{final} \omega_{final} \]
Lesi simiso sibaluleke kakhulu ezimweni ezahlukahlukene zomzimba, njengokunyakaza kwamaplanethi, ama-pirouette abadansi, kanye nokuzinza kwama-gyroscope.
Ukusetshenziswa kwe-Angular Momentum Empilweni Yansuku Zonke
Ukuhamba Kweplanethi
Amaplanethi ohlelweni lwelanga azungeza ilanga futhi anomfutho ojikelezayo ocishe ufane. Izinguquko ezincane kumfutho ojikelezayo zingabangela izinguquko ekujikelezeni kweplanethi. Lokhu kungenxa yokuthi amandla adonsela phansi asebenza kule planethi awakhiqizi torque eqondile, okugcina umfutho ojikelezayo ungaguquki.
Umdansi weBallet uPirouette
Umdansi we-ballet angandisa isivinini sokujikeleza kwakhe ngokudonsa izingalo nemilenze yakhe eduze komzimba wakhe. Lokhu kungenxa yokuthi umzuzu wokungalali kahle uyancipha, ngakho-ke ukuze kulondolozwe umfutho we-angular oqhubekayo, ijubane le-angular kumele lande.
I-Gyroscope
I-gyroscope iyithuluzi elisebenzisa isimiso sokunyakaza kwe-angular ukuze ligcine ukuzinza. Ama-gyroscope asetshenziswa ezinhlotsheni ezahlukene zokusebenza, njengokuzulazula kwezindiza, imikhumbi, kanye ne-smartphone.
Imibuzo nezixazululo zezibonelo
Isibonelo Umbuzo 1
Idiski enesisindo esingu-2 kg kanye nerediyasi yamamitha angu-0,5 ijikeleza ngejubane eliyi-angle elingu-10 rad/s. Bala umfutho we-angle wediski.
Isixazululo:
Isikhathi sokungabi namandla kwediski (\(I\)) sinikezwa yifomula:
\[ I = \frac{1}{2} mr^2 \]
Faka amanani anikeziwe:
\[ I = \frac{1}{2} \izikhathi 2 \, \text{kg} \izikhathi (0,5 \, \text{m})^2 = \frac{1}{2} \izikhathi 2 \izikhathi 0,25 = 0,25 \, \text{kg} \cdot \text{m}^2 \]
Umfutho we-angular (\(L\)) ngu:
\[ L = I \omega = 0,25 \, \text{kg} \cdot \text{m}^2 \times 10 \, \text{rad/s} = 2,5 \, \text{kg} \cdot \text{m}^2/\text{s} \]
Isibonelo Umbuzo 2
Umskeyiti one-inertia yokuqala engu-0,8 kg·m² ujikeleza ngejubane eliyi-angle elingu-5 rad/s. Uma ehoxisa izingalo zakhe futhi i-inertia yakhe yehla ibe ngu-0,4 kg·m², iyini ijubane lakhe lokugcina eliyi-angle?
Isixazululo:
Ukusebenzisa isimiso sokulondolozwa komfutho we-angular:
\[ I_{initial} \omega_{initial} = I_{final} \omega_{final} \]
Faka amanani anikeziwe:
\[ 0,8 \, \text{kg} \cdot \text{m}^2 \times 5 \, \text{rad/s} = 0,4 \, \text{kg} \cdot \text{m}^2 \times \omega_{end} \]
\[ 4 \, \text{kg} \cdot \text{m}^2/\text{s} = 0,4 \, \text{kg} \cdot \text{m}^2 \times \omega_{end} \]
\[ \omega_{end} = \frac{4 \, \text{kg} \cdot \text{m}^2/\text{s}}{0,4 \, \text{kg} \cdot \text{m}^2} = 10 \, \text{rad/s} \]
Isiphetho
Umfutho we-angular ngumqondo obalulekile ohlobene nokunyakaza kwezinto okuzungezayo. Amafomula ayisisekelo omfutho we-angular, \(\vec{L} = \vec{r} \times \vec{p}\) kanye \(L = I \omega\), anikeza isisekelo sokuqonda izinhlobo eziningi zezehlakalo ezibonakalayo. Isimiso sokulondolozwa komfutho we-angular sisiza ukuchaza nokubikezela ukuziphatha kwezinhlelo ezizungezayo ezimweni eziningi, kusukela ekunyakazeni kweplanethi kuya ku-ballet. Ngokuqonda umqondo kanye nokusetshenziswa komfutho we-angular, singabuqonda kangcono ubuhle kanye nobunzima bokunyakaza okuzungezayo endaweni yonke.
