Maitiro Ekuverenga Angular Momentum
Angular momentum ipfungwa yakakosha mufizikisi, kunyanya mu classical uye quantum mechanics. Muchinyorwa chino, tichakurukura zvakadzama maverengero e angular momentum, nzira dzakasiyana-siyana dziripo, uye mashandisirwo ayo muhupenyu hwezuva nezuva. Kunzwisisa pfungwa iyi kunobatsira kwete chete kune vadzidzi vefizikisi nenyanzvi, asiwo kune chero ani zvake anofarira mashandiro ezvisikwa padanho rekutanga.
Pendauluan
Angular momentum inhamba yevector inotsanangura kutenderera kwechinhu chakatenderedza poindi. Sezvo linear momentum iine chekuita nekufamba kwakarongeka, angular momentum inodzora kuti chinhu chinotenderera sei. Fomura yekutanga ye angular momentum (\(L\)) ichibereko chenguva yekusagadzikana (\(I\)) uye angular velocity (\(\omega\)):
\[ L = I \cdot \omega \]
Zvisinei, kana tikafunga nezvenyaya yechinhu chinofamba-famba panzvimbo, fomura inoshandiswa ndeiyi:
\[ L = r \nguva p \]
Di mana:
– \( r \) ndiyo vhekitari yenzvimbo yechinhu ichi maererano nepakati pekutenderera.
– \( p \) ndiyo linear momentum ye particle (\( p = m \cdot v \) apo \( m \) iri huremu hwe particle uye \( v \) ndiyo linear velocity).
Chiratidzo "\(\times\)" chinomiririra muchinjikwa wezvipeki, zvinoreva kuti angular momentum inogara yakatarisana nendege inoumbwa ne position vector \( r \) uye momentum vector \( p \).
Kuverenga Angular Momentum muDiscrete Systems
Ngatitii tine chinhu chine huremu \( m \) chinofamba nekumhanya \( v \) chiri kure \( r \) kubva pakati pekutenderera. Matanho ekuverenga angular momentum ndeaya anotevera:
1. Sarudza Position Vector (\( r \)) uye Momentum Vector (\( p \)):
Ita shuwa kuti mavector ese anoyerwa kubva pakati pekutenderera. Ngatitii particle iri panzvimbo \( (x, y, z) \) uye ichifamba nekumhanya \( (v_x, v_y, v_z) \). Zvadaro, position vector ndi \( \vec{r} = (x, y, z) \), uye momentum vector ndi \( \vec{p} = m \cdot (v_x, v_y, v_z) \).
2. Verenga Chigadzirwa Chinochinjika (\( \vec{r} \times \vec{p} \)):
Chibereko chekubatanidza mavector maviri muCartesian coordinates chinogona kuverengerwa ne:
\[
\vec{L} = \vec{r} \times \vec{p} = \left( \begin{array}{c}
y \cdot p_z – z \cdot p_y \\
z \cdot p_x – x \cdot p_z \\
x \cdot p_y – y \cdot p_x \\
\end{array} \right)
\]
3. Kuongorora Kukosha uye Kutungamirirwa kweAngular Momentum:
Mhedzisiro yechigadzirwa chakachinjika ivhekitari ine gwara rakati uye hukuru. Hukuru hwe angular momentum hunogona kuverengerwa nekutora hukuru hwevhekitari \(\vec{L}\):
\[
|\vec{L}| = \sqrt{(L_x)^2 + (L_y)^2 + (L_z)^2}
\]
Kuverenga Angular Momentum muContinuous Systems
Kune zvinhu zvine kugoverwa kwehuwandu kunoramba kuripo, senge tsvimbo inotenderera kana dhisiki, matanho akajairika ndeaya anotevera:
1. Sarudza Nguva Yekusagadzikana (\( I \)):
Nguva yekusagadzikana itensor inotsanangura kuti huremu hwechinhu hunogoverwa sei zvichienderana ne axis of rotation yacho. Mimwe mienzaniso yenguva dzekusagadzikana kwechimiro chechinhu chakasiyana:
– Tsvimbo refu \( L \) ine kutenderera pakati: \( I = \frac{1}{12} m L^2 \)
– Dhisiki ine radius \( R \): \( I = \frac{1}{2} m R^2 \)
– Dunhu rakasimba rine radius \( R \): \( I = \frac{2}{5} m R^2 \)
2. Sarudza Angular Velocity (\( \omega \)):
Kumhanya kweAngular ndiko kukurumidza kunoita chinhu kutenderera uye kunowanzo kuyerwa mu radians pasekondi.
3. Wedzera Nguva yeKusasimba neAngular Velocity:
Shandisa fomura \( L = I \cdot \omega \) kuti uwane angular momentum yechinhu.
Muenzaniso wezvinetso
Muenzaniso 1: Zvidimbu Zvinofamba Mumutsara Wakatwasuka
Ngatitii chidimbu chine huremu hwe2 kg chinofamba nekumhanya kwe3 m/s munzira \( \hat{i} \) uye chiri panzvimbo yemamita maviri kubva pa axis yekutenderera munzira \( \hat{j} \).
1. Vekitari yenzvimbo \( \vec{r} = 2 \hat{j} \)
2. Vekitari yekukurumidza \( \vec{p} = 2 \kawa 3 \hat{i} = 6 \hat{i} \)
3. Chigadzirwa chakasiyana \( \vec{L} = \vec{r} \times \vec{p} \):
\[
\vec{L} = \begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
0 & 2 & 0 \\
6 & 0 & 0 \\
\end{vmatrix} = (0)(0) – (2)(0) \hat{i} – (0)(0) + (6)(0) \hat{j} + (2)(6) – (0)(0) \hat{k}
= (0 \hat{i}, -0 \hat{j}, 12 \hat{k})
= 12 \hat{k}
\]
Saka, \( \vec{L} = 12 \hat{k} \, \text{kg} \cdot \text{m}^2 / \text{s} \).
Muenzaniso 2: Dhisiki Inotenderera
Dhisiki rakafanana rine huremu hwe5 kg uye radius ye 0.5 metres rinotenderera ne angular velocity ye 10 radians/s.
1. Nguva yekusaita basa, \( I = \frac{1}{2} m R^2 = \frac{1}{2} \times 5 \times (0.5)^2 = \frac{1}{2} \times 5 \times 0.25 = 0.625 \, \text{kg} \cdot \text{m}^2 \)
2. Kumhanya kwekona, \( \omega = 10 \, \text{rad/s} \)
3. Angular momentum, \( L = I \cdot \omega = 0.625 \times 10 = 6.25 \, \text{kg} \cdot \text{m}^2 / \text{s} \)
Kushandiswa kweAngular Momentum
Kunzwisisa angular momentum kune mashandisirwo akasiyana-siyana anoshanda. Semuenzaniso:
– Astrophysics: Simba rinokweva nyeredzi iri kufa rinoita kuti mapuraneti akaipoteredza arambe achienderera mberi nekufamba kwawo, izvo zvine chekuita nekutenderera kwawo achitenderera nyeredzi.
– Simba reMhepo: Maturbine emhepo anoshandisa musimboti we angular momentum kushandura simba remhepo kuita simba remagetsi.
– Mitambo: Vatambi vanowanzo shandisa musimboti we angular momentum mukufamba kwakasiyana-siyana, sekutenderera mukunyura kana kukanda pfumo.
Mhedziso
Angular momentum ipfungwa yakadzama uye inoshanda mufizikisi. Nekunzwisisa maitiro ekuiverenga kune ese ari maviri masisitimu akaparadzana uye anoenderera mberi, tinogona kuwana kunzwisisa kwakajeka kwekutenderera uye kuenzana kwezvinhu zvakasiyana-siyana. Mabhenefiti eruzivo urwu anodarika dzidzo kusvika kumashandisirwo anoshanda muhupenyu hwezuva nezuva.
