Haʻilula Momentum Angular
Pendahuluan
He manaʻo koʻikoʻi ka momentum angular i ka physics e pili ana i ka neʻe ʻana o kahi mea. Ua like kēia manaʻo me ka momentum linear i ka neʻe ʻana o ka neʻe ʻana. He kuleana koʻikoʻi ka momentum angular i nā ʻano like ʻole o ka physics, mai ka mechanics kuʻuna a i ka mechanics quantum. E kūkākūkā kēia ʻatikala i ka wehewehe ʻana o ka momentum angular, nā ʻano hana pili, nā noi i ke ola o kēlā me kēia lā, a me nā laʻana e hoʻoikaika i ka ʻike.
Ka Wehewehena o ka Momentum Angular
ʻO ka momentum angular kahi nui vector e wehewehe ana i ke ʻano o kahi mea e hoʻomau i ka wili ʻana a puni kahi kiko a i ʻole axis. Pili ka momentum angular (\(\vec{L}\)) i ʻelua mau kumu nui: momentum linear (\(\vec{p}\)) a me ke kūlana pili (\(\vec{r}\)) o kahi kiko kuhikuhi. Ua wehewehe ʻia ka momentum angular penei:
\[ \vec{L} = \vec{r} \times \vec{p} \]
Ma hea:
– ʻO \(\vec{L}\) ka momentum angular.
– ʻO \(\vec{r}\) ka vector kūlana e pili ana i ke kiko kuhikuhi.
– ʻO \(\vec{p}\) ka momentum linear (\(\vec{p} = m \vec{v}\), kahi ʻo \(m\) ka nuipa a ʻo \(\vec{v}\) ka wikiwiki).
– Hōʻike ʻo \(\times\) i ka huahana kea ma waena o nā vectors ʻelua.
Haʻilula Momentum Angular
No ke kino paʻa e wili ana me ka wikiwiki angular (\(\omega\)) e pili ana i kahi axis paʻa, hiki ke hōʻike ʻia ka momentum angular (\(L\)) penei:
\[ L = I \omega \]
Ma hea:
– ʻO \(L\) ka momentum angular.
– ʻO \(I\) ka manawa o ka inertia o ka mea e pili ana i ke axis o ka wili.
– ʻO \(\omega\) ka wikiwiki kihi.
Manawa o ka Inertia
ʻO ke ana o ka manawa o ka inertia (\(I\)) he ana ia o ke kū'ē ʻana o kahi mea i nā loli o kāna neʻe wili. Aia ke ana o ka manawa o ka inertia i ka hoʻolaha ʻana o ka nuipa o ka mea e pili ana i ke axis o ka wili. No kahi mea paʻa, hiki ke helu ʻia ke ana o ka manawa o ka inertia me ka hoʻohana ʻana i ke ʻano hana:
\[ I = \sum m_i r_i^2 \]
Ma hea:
– ʻO \(m_i\) ka nuipa o ka ʻāpana \(i\)th.
– ʻO \(r_i\) ka mamao o ka ʻāpana \(i\)th mai ke axis o ka wili ʻana.
No nā mea maʻalahi, loaʻa i ka manawa o ka inertia kāna ʻano hana ponoʻī. Eia kekahi mau laʻana:
– Paukū hakahaka: \(I = mr^2\)
– Paukū piha: \(I = \frac{1}{2} mr^2\)
– Pōʻai Piha: \(I = \frac{2}{5} mr^2\)
Kumumanaʻo o ka Mālama ʻana i ka Momentum Angular
ʻO ke kumumanaʻo o ka mālama ʻana i ka momentum angular e ʻōlelo ana inā ʻaʻohe torque waho e hana ma kahi ʻōnaehana, e mau ana ka momentum angular holoʻokoʻa o ka ʻōnaehana. ʻO ke ʻano kēia:
\[ \vec{L}_{hoʻomaka} = \vec{L}_{hope} \]
a i ʻole
\[ I_{mua} \omega_{mua} = I_{hope} \omega_{hope} \]
He mea nui loa kēia kumumanaʻo i nā hanana kino like ʻole, e like me ka neʻe ʻana o nā hōkūhele, nā pirouettes o ka poʻe hula, a me ke kūpaʻa o nā gyroscopes.
Ka Hoʻohana ʻana o ka Momentum Angular i ke Ola o kēlā me kēia lā
Neʻe Honua
Hoʻopuni nā hōkūhele o ka ʻōnaehana lā a puni ka lā a loaʻa iā lākou ka momentum angular kokoke mau. Hiki i nā loli liʻiliʻi i ka momentum angular ke hoʻololi i ka orbit o ka honua. ʻO kēia no ka mea ʻaʻole hana ka ikaika umekaumaha e hana ana ma ka honua i ka torque net, e mālama ana i ka momentum angular mau.
Ka mea hula ballet Pirouette
Hiki i ka mea hula ballet ke hoʻonui i ka wikiwiki o kāna wili ʻana ma ka huki ʻana i kona mau lima a me nā wāwae a kokoke i kona kino. ʻO kēia no ka mea e emi ana ka manawa o ka inertia, no laila e mālama i ka momentum angular mau, pono e hoʻonui i ka velocity angular.
ʻO ke Gyroscope
He mea hana ka gyroscope e hoʻohana ana i ke kumumanaʻo o ka momentum angular e mālama i ke kūpaʻa. Hoʻohana ʻia nā gyroscopes i nā ʻano hana like ʻole, e like me ka mokulele, ka moku, a me ka hoʻokele kelepona akamai.
Nā nīnau a me nā pane hoʻohālike
Laʻana Nīnau 1
Hoʻohuli ʻia kahi diski me ke kaumaha o 2 kg a me ka radius o 0,5 mika ma ka wikiwiki angular o 10 rad/s. E helu i ka momentum angular o ka diski.
Hoʻonā:
Ua hāʻawi ʻia ka manawa o ka inertia o ka diski (\(I\)) e ke ʻano hana:
\[ I = \frac{1}{2} mr^2 \]
E hoʻokomo i nā waiwai i hāʻawi ʻia:
\[ I = \frac{1}{2} \times 2 \, \text{kg} \times (0,5 \, \text{m})^2 = \frac{1}{2} \times 2 \times 0,25 = 0,25 \, \text{kg} \cdot \text{m}^2 \]
ʻO ka momentum angular (\(L\)):
\[ L = I \omega = 0,25 \, \text{kg} \cdot \text{m}^2 \times 10 \, \text{rad/s} = 2,5 \, \text{kg} \cdot \text{m}^2/\text{s} \]
Laʻana Nīnau 2
Ke milo nei kekahi mea heʻe hau me ka manawa inertia mua o 0,8 kg·m² ma ka wikiwiki angular o 5 rad/s. Inā hoʻihoʻi ʻo ia i kona mau lima a emi iho kona manawa inertia i 0,4 kg·m², he aha kona wikiwiki angular hope loa?
Hoʻonā:
Ke hoʻohana nei i ke kumumanaʻo o ka mālama ʻana i ka momentum angular:
\[ I_{mua} \omega_{mua} = I_{hope} \omega_{hope} \]
E hoʻokomo i nā waiwai i hāʻawi ʻia:
\[ 0,8 \, \text{kg} \cdot \text{m}^2 \times 5 \, \text{rad/s} = 0,4 \, \text{kg} \cdot \text{m}^2 \times \omega_{end} \]
\[ 4 \, \text{kg} \cdot \text{m}^2/\text{s} = 0,4 \, \text{kg} \cdot \text{m}^2 \times \omega_{end} \]
\[ \omega_{end} = \frac{4 \, \text{kg} \cdot \text{m}^2/\text{s}}{0,4 \, \text{kg} \cdot \text{m}^2} = 10 \, \text{rad/s} \]
Ka hopena
He manaʻo koʻikoʻi ka momentum angular e pili ana i ka neʻe ʻana o nā mea i ka wili ʻana. ʻO nā ʻano kumu no ka momentum angular, \(\vec{L} = \vec{r} \times \vec{p}\) a me \(L = I \omega\), hāʻawi i ke kumu no ka hoʻomaopopo ʻana i nā ʻano hanana kino like ʻole. Kōkua ke kumumanaʻo o ka mālama ʻana i ka momentum angular i ka wehewehe a me ka wānana i ke ʻano o nā ʻōnaehana wili i nā kūlana he nui, mai ka neʻe ʻana o ka honua a i ka ballet. Ma ka hoʻomaopopo ʻana i ke kumumanaʻo a me nā noi o ka momentum angular, hiki iā mākou ke mahalo maikaʻi i ka nani a me ka paʻakikī o ka neʻe ʻana o ka wili ma ke ao holoʻokoʻa.
