Momwe Mungawerengere Mphamvu ya Angular
Angular momentum ndi lingaliro lofunika kwambiri mu fizikisi, makamaka mu makina akale ndi a quantum. M'nkhaniyi, tikambirana mwatsatanetsatane momwe tingawerengere angular momentum, njira zosiyanasiyana zomwe zilipo, ndi momwe imagwiritsidwira ntchito pa moyo watsiku ndi tsiku. Kumvetsetsa lingaliro ili ndikopindulitsa osati kwa ophunzira ndi akatswiri a fizikisi okha, komanso kwa aliyense amene akufuna kudziwa momwe chilengedwe chimagwirira ntchito pamlingo woyambira.
Pendauluan
Mphamvu ya angular ndi kuchuluka kwa vector komwe kumafotokoza kuzungulira kwa chinthu mozungulira mfundo. Monga momwe mphamvu ya mzere imagwirizanirana ndi kuyenda kolunjika, mphamvu ya angular imalamulira momwe chinthu chimazungulira. Fomula yoyambira ya mphamvu ya angular (\(L\)) ndi zotsatira za mphindi ya inertia (\(I\)) ndi liwiro la angular (\(\omega\)):
\[ L = I \cdot \omega \]
Komabe, ngati tiganizira za tinthu tomwe tikuyenda mozungulira mfundo, njira yomwe imagwiritsidwa ntchito ndi iyi:
\[ L = r \times p \]
Kumene:
– \( r \) ndi vekitala ya malo a tinthu tomwe timakhala pakati pa kuzungulira.
– \( p \) ndi mzere wa tinthu tating'onoting'ono (\( p = m \cdot v \) pomwe \( m \) ndi kulemera kwa tinthu tating'onoting'ono ndipo \( v \) ndi mzere wa liwiro).
Chizindikiro "\(\times\)" chikuyimira zotsatira za ma vector, zomwe zikutanthauza kuti angular momentum nthawi zonse imakhala yolunjika ku ndege yopangidwa ndi position vector \( r \) ndi momentum vector \( p \).
Kuwerengera Angular Momentum mu Discrete Systems
Tiyerekeze kuti tili ndi tinthu tomwe timakhala ndi kulemera \( m \) tikuyenda ndi liwiro \( v \) patali \( r \) kuchokera pakati pa kuzungulira. Masitepe owerengera angular momentum ndi awa:
1. Dziwani Malo Vekitala (\( r \)) ndi Momentum Vekitala (\( p \)):
Onetsetsani kuti ma vector onse ayesedwa kuchokera pakati pa kuzungulira. Tiyerekeze kuti tinthu tating'onoting'ono tili pamalo \( (x, y, z) \) ndipo tikuyenda ndi liwiro \( (v_x, v_y, v_z) \). Kenako, vector ya malo ndi \( \vec{r} = (x, y, z) \), ndipo vector ya momentum ndi \( \vec{p} = m \cdot (v_x, v_y, v_z) \).
2. Werengani Chogulitsa Chosiyanasiyana (\( \vec{r} \times \vec{p} \)):
Chogulitsa chophatikizana cha ma vector awiri mu ma coordinates a Cartesian chikhoza kuwerengedwa ndi:
\[
\vec{L} = \vec{r} \times \vec{p} = \left( \begin{array}{c}
y \cdot p_z – z \cdot p_y \\
z \cdot p_x – x \cdot p_z \\
x \cdot p_y – y \cdot p_x \\
\end{array} \right)
\]
3. Kuwunika Mtengo ndi Malangizo a Angular Momentum:
Zotsatira za chinthu chopingasa ndi vekitala yokhala ndi njira ndi kukula kwake. Kukula kwa angular momentum kumatha kuwerengedwa potengera kukula kwa vekitala \(\vec{L}\):
\[
|\vec{L}| = \sqrt{(L_x)^2 + (L_y)^2 + (L_z)^2}
\]
Kuwerengera Mphamvu ya Angular mu Machitidwe Opitilira
Pa zinthu zomwe zimakhala ndi kugawa kosalekeza kwa kulemera, monga ndodo yozungulira kapena diski, njira zambiri ndi izi:
1. Dziwani Nthawi ya Kusakhazikika (\( I \)):
Nthawi ya inertia ndi tensor yomwe imafotokoza momwe kulemera kwa chinthu kumagawidwira poyerekeza ndi mzere wake wozungulira. Zitsanzo zina za nthawi ya inertia ya mawonekedwe osiyanasiyana a chinthu:
– Ndodo yayitali \( L \) yokhala ndi kuzungulira pakati: \( I = \frac{1}{12} m L^2 \)
– Disiki yokhala ndi radius \( R \): \( I = \frac{1}{2} m R^2 \)
– Chipilala cholimba chokhala ndi radius \( R \): \( I = \frac{2}{5} m R^2 \)
2. Dziwani Kuthamanga kwa Angular (\( \omega \)):
Liwiro la angular ndi momwe chinthu chimazungulira mofulumira ndipo nthawi zambiri chimayesedwa mu ma radians pa sekondi.
3. Chulukitsani Nthawi ya Kusakhazikika ndi Angular Velocity:
Gwiritsani ntchito fomula \( L = I \cdot \omega \) kuti mupeze angular momentum ya chinthucho.
Chitsanzo cha mavuto
Chitsanzo 1: Tinthu Timene Timayenda Mzere Wowongoka
Tiyerekeze kuti tinthu tolemera makilogalamu awiri tikuyenda pa liwiro la 3 m/s molunjika \( \hat{i} \) ndipo tili pamalo a mamita awiri kuchokera ku mzere wozungulira molunjika \( \hat{j} \).
1. Vekitala ya malo \( \vec{r} = 2 \hat{j} \)
2. Vekitala ya Momentum \( \vec{p} = 2 \times 3 \hat{i} = 6 \hat{i} \)
3. Chogulitsa chosakanikirana \( \vec{L} = \vec{r} \times \vec{p} \):
\[
\vec{L} = \begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
0 & 2 & 0 \\
6 & 0 & 0 \\
\end{vmatrix} = (0)(0) – (2)(0) \hat{i} – (0)(0) + (6)(0) \hat{j} + (2)(6) – (0)(0) \hat{k}
= (0 \hat{i}, -0 \hat{j}, 12 \hat{k})
= 12 \hat{k}
\]
Kotero, \( \vec{L} = 12 \hat{k} \, \text{kg} \cdot \text{m}^2 / \text{s} \).
Chitsanzo 2: Disiki Yozungulira
Disiki yofanana yokhala ndi kulemera kwa 5 kg ndi utali wa mamita 0.5 imazungulira ndi liwiro la angular la ma radian 10/s.
1. Nthawi ya kulephera, \( I = \frac{1}{2} m R^2 = \frac{1}{2} \times 5 \times (0.5)^2 = \frac{1}{2} \times 5 \times 0.25 = 0.625 \, \text{kg} \cdot \text{m}^2 \)
2. Liwiro la angular, \( \omega = 10 \, \text{rad/s} \)
3. Angular momentum, \( L = I \cdot \omega = 0.625 \times 10 = 6.25 \, \text{kg} \cdot \text{m}^2 / \text{s} \)
Kugwiritsa Ntchito Angular Momentum
Kumvetsetsa mphamvu ya angular kuli ndi ntchito zosiyanasiyana zothandiza. Mwachitsanzo:
– Astrophysics: Mphamvu yokoka ya nyenyezi yomwe ikufa imapangitsa mapulaneti ozungulira nyenyeziyo kusunga mphamvu yawo yozungulira, zomwe zimakhudza kuzungulira kwawo mozungulira nyenyeziyo.
– Mphamvu ya Mphepo: Ma turbine a mphepo amagwiritsa ntchito mfundo ya angular momentum kuti asinthe mphamvu ya kinetic ya mphepo kukhala mphamvu yamagetsi.
– Masewera: Othamanga nthawi zambiri amagwiritsa ntchito mfundo ya angular momentum m'mayendedwe osiyanasiyana, monga kuzungulira podumphira m'madzi kapena kuponya mikondo.
Mapeto
Mphamvu ya Angular ndi lingaliro lozama komanso lothandiza mu fizikisi. Mwa kumvetsetsa momwe tingawerengere ma discrete ndi continuous systems, titha kumvetsetsa bwino za kuzungulira ndi kufanana kwa zinthu zosiyanasiyana. Ubwino wa chidziwitsochi umapitilira kupitirira maphunziro mpaka kugwiritsidwa ntchito m'moyo watsiku ndi tsiku.
