Fomula Yoyenda Mofulumira ya Angular
Pendauluan
Mphamvu ya angular ndi lingaliro lofunika kwambiri mu fizikisi yokhudzana ndi kayendedwe ka chinthu. Lingaliroli likufanana ndi mphamvu yolunjika mu kayendedwe ka kumasulira. Mphamvu ya angular imagwira ntchito yofunika kwambiri m'magawo osiyanasiyana a fizikisi, kuyambira makina akale mpaka makina a quantum. Nkhaniyi ikambirana tanthauzo la mphamvu ya angular, ma formula okhudzana nawo, ntchito m'moyo watsiku ndi tsiku, ndi zitsanzo zokulitsa kumvetsetsa.
Tanthauzo la Angular Momentum
Mphamvu ya angular ndi kuchuluka kwa vector komwe kumafotokoza momwe chinthu chimakhalira chozungulira mozungulira mfundo kapena mzere. Mphamvu ya angular (\(\vec{L}\)) imadalira zinthu ziwiri zazikulu: mphamvu yolunjika (\(\vec{p}\)) ndi malo ogwirizana (\(\vec{r}\)) a mfundo yofotokozera. Mphamvu ya angular imatanthauzidwa motere:
\[ \vec{L} = \vec{r} \times \vec{p} \]
Kumene:
– \(\vec{L}\) ndi angular momentum.
– \(\vec{r}\) ndi vekitala ya malo poyerekeza ndi malo ofotokozera.
– \(\vec{p}\) ndi mzere wa mphamvu (\(\vec{p} = m \vec{v}\), pomwe \(m\) ndi kulemera ndipo \(\vec{v}\) ndi liwiro).
– \(\times\) ikuyimira chinthu chosakanikirana pakati pa mavekitala awiri.
Fomula Yoyenda Mofulumira ya Angular
Pa thupi lolimba lomwe likuzungulira ndi liwiro la angular (\(\omega\)) pafupi ndi mzere wokhazikika, mphamvu ya angular (\(L\)) ikhoza kufotokozedwa motere:
\[ L = I \omega \]
Kumene:
– \(L\) ndi mphamvu ya angular.
– \(I\) ndi nthawi ya inertia ya chinthu chozungulira mzere wozungulira.
– \(\omega\) ndi liwiro la angular.
Nthawi ya Inertia
Nthawi ya inertia (\(I\)) ndi muyeso wa kukana kwa chinthu ku kusintha kwa kayendedwe kake kozungulira. Nthawi ya inertia imadalira kugawa kwa kulemera kwa chinthucho poyerekeza ndi mzere wozungulira. Pa chinthu cholimba, nthawi ya inertia ikhoza kuwerengedwa pogwiritsa ntchito fomula iyi:
\[ I = \sum m_i r_i^2 \]
Kumene:
– \(m_i\) ndi kulemera kwa tinthu ta \(i\)th.
– \(r_i\) ndi mtunda wa tinthu ta \(i\)th kuchokera ku mzere wozungulira.
Pa zinthu zosavuta, nthawi ya inertia ili ndi njira yakeyake. Zitsanzo zina ndi izi:
– Silinda Yopanda Pang'ono: \(I = mr^2\)
– Silinda Yonse: \(I = \frac{1}{2} mr^2\)
– Full Sphere: \(I = \frac{2}{5} mr^2\)
Mfundo Yosungira Mphamvu Yozungulira
Mfundo yosungira mphamvu ya angular imati ngati palibe mphamvu yakunja yomwe ikugwira ntchito pa dongosolo, mphamvu yonse ya angular ya dongosololi idzakhalabe yofanana. Izi zikutanthauza:
\[ \vec{L}_{start} = \vec{L}_{end} \]
kapena
\[ I_{initial} \omega_{initial} = I_{final} \omega_{final} \]
Mfundo imeneyi ndi yofunika kwambiri pa zochitika zosiyanasiyana zakuthupi, monga kuyenda kwa mapulaneti, ma pirouette a ovina, ndi kukhazikika kwa ma gyroscope.
Kugwiritsa Ntchito Mphamvu ya Angular mu Moyo wa Tsiku ndi Tsiku
Kuyenda kwa Mapulaneti
Mapulaneti omwe ali mu dongosolo la dzuwa amazungulira dzuwa ndipo amakhala ndi mphamvu yozungulira pafupifupi nthawi zonse. Kusintha pang'ono kwa mphamvu yozungulira kungayambitse kusintha kwa kayendedwe ka dziko lapansi. Izi zili choncho chifukwa mphamvu yokoka yomwe imagwira ntchito padziko lapansi siimapanga mphamvu yozungulira, zomwe zimapangitsa kuti mphamvu yozungulira ikhale yofanana.
Wovina wa Ballet Pirouette
Wovina ballet amatha kuwonjezera liwiro la kuzungulira kwake mwa kukoka manja ndi miyendo yake pafupi ndi thupi lake. Izi zili choncho chifukwa nthawi ya kulephera kugwira ntchito imachepa, kotero kuti pakhale mphamvu yokhazikika ya angular, liwiro la angular liyenera kuwonjezeka.
Gyroscope
Gyroscope ndi chipangizo chomwe chimagwiritsa ntchito mfundo ya angular momentum kuti chikhale chokhazikika. Ma Gyroscope amagwiritsidwa ntchito m'njira zosiyanasiyana, monga kuyendetsa ndege, sitima, ndi mafoni.
Zitsanzo za mafunso ndi mayankho
Chitsanzo cha Funso 1
Disiki yokhala ndi kulemera kwa 2 kg ndi utali wa mamita 0,5 imazungulira pa liwiro la angular la 10 rad/s. Werengani mphamvu ya angular ya disiki.
Yankho:
Nthawi ya inertia ya disc (\(I\)) imaperekedwa ndi fomula:
\[ I = \frac{1}{2} mr^2 \]
Lowetsani mfundo zomwe zaperekedwa:
\[ I = \frac{1}{2} \nthawi 2 \, \text{kg} \nthawi (0,5 \, \text{m})^2 = \frac{1}{2} \nthawi 2 \nthawi 0,25 = 0,25 \, \text{kg} \cdot \text{m}^2 \]
Mphamvu ya angular (\(L\)) ndi:
\[ L = I \omega = 0,25 \, \text{kg} \cdot \text{m}^2 \times 10 \, \text{rad/s} = 2,5 \, \text{kg} \cdot \text{m}^2/\text{s} \]
Chitsanzo cha Funso 2
Wosewera skate wokhala ndi nthawi yoyambira ya inertia ya 0,8 kg·m² akuzungulira pa liwiro la angular la 5 rad/s. Ngati abweza manja ake ndipo nthawi yake ya inertia ikuchepa kufika pa 0,4 kg·m², kodi liwiro lake lomaliza la angular ndi lotani?
Yankho:
Pogwiritsa ntchito mfundo yosungira mphamvu ya angular:
\[ I_{initial} \omega_{initial} = I_{final} \omega_{final} \]
Lowetsani mfundo zomwe zaperekedwa:
\[ 0,8 \, \text{kg} \cdot \text{m}^2 \times 5 \, \text{rad/s} = 0,4 \, \text{kg} \cdot \text{m}^2 \times \omega_{end} \]
\[ 4 \, \text{kg} \cdot \text{m}^2/\text{s} = 0,4 \, \text{kg} \cdot \text{m}^2 \times \omega_{end} \]
\[ \omega_{end} = \frac{4 \, \text{kg} \cdot \text{m}^2/\text{s}}{0,4 \, \text{kg} \cdot \text{m}^2} = 10 \, \text{rad/s} \]
Mapeto
Mphamvu ya angular ndi lingaliro lofunika kwambiri lokhudzana ndi kayendedwe ka zinthu. Ma formula oyambira a mphamvu ya angular, \(\vec{L} = \vec{r} \times \vec{p}\) ndi \(L = I \omega\), amapereka maziko omvetsetsa zochitika zosiyanasiyana zakuthupi. Mfundo yosungira mphamvu ya angular imathandiza kufotokoza ndi kulosera machitidwe ozungulira m'mikhalidwe yambiri, kuyambira kuyenda kwa mapulaneti mpaka ku ballet. Mwa kumvetsetsa lingaliro ndi kugwiritsa ntchito mphamvu ya angular, titha kuyamikira bwino kukongola ndi zovuta za kayendedwe ka kuzungulira m'chilengedwe chonse.
