Yadda Ake Lissafin Ƙarfin Angular Momentum
Motsin kusurwa muhimmin ra'ayi ne a fannin kimiyyar lissafi, musamman a fannin kimiyyar lissafi da kuma kimiyyar lissafi. A cikin wannan labarin, za mu tattauna dalla-dalla kan yadda ake ƙididdige motsin kusurwa, hanyoyi daban-daban da ake da su, da kuma aikace-aikacensu a rayuwar yau da kullum. Fahimtar wannan ra'ayi yana da amfani ba kawai ga ɗaliban kimiyyar lissafi da ƙwararru ba, har ma ga duk wanda ke sha'awar yadda yanayi ke aiki a matakin asali.
Pendahuluan
Motsin kusurwa wani adadi ne na vector wanda ke bayyana juyawar abu a kusa da wani wuri. Kamar yadda motsin layi ke da alaƙa da motsi na layi, motsin kusurwa yana sarrafa yadda abu ke juyawa. Tsarin asali na motsin kusurwa (\(L\)) shine samfurin lokacin inertia (\(I\)) da saurin kusurwa (\(\omega\)):
\[ L = I \cdot \omega \]
Duk da haka, idan muka yi la'akari da yanayin ƙwayar cuta da ke motsawa a kusa da wani wuri, dabarar da aka yi amfani da ita ita ce:
\[ L = r \times p \]
Ina:
– \( r \) shine siginar matsayi na ƙwayar cuta dangane da tsakiyar juyawa.
– \( p \) shine motsi mai layi na ƙwayar cuta (\( p = m \cdot v \) inda \( m \) shine nauyin ƙwayar cuta kuma \( v \) shine saurin layi).
Alamar "\(\times\)" tana wakiltar samfurin giciye na vectors, wanda ke nufin cewa ƙarfin kusurwa koyaushe yana tsaye ne da jirgin sama wanda vector position \( r \) da vector momentum \( p \) suka samar.
Lissafin Ƙarfin Kusurwa a Tsarin Na Musamman
A ce muna da barbashi mai nauyin \( m \) yana motsawa tare da saurin \( v \) a nesa \( r \) daga tsakiyar juyawa. Matakan da za a bi don ƙididdige motsin kusurwa sune kamar haka:
1. Ƙayyade Matsayin Vector (\( r \)) da Momentum Vector (\( p \)):
Tabbatar an auna dukkan vectors daga tsakiyar juyawa. A ce barbashi yana kan matsayi \( (x, y, z) \) kuma yana motsawa tare da saurin \( (v_x, v_y, v_z) \). Sannan, vector ɗin matsayi shine \( \vec{r} = (x, y, z) \), kuma vector ɗin motsi shine \( \vec{p} = m \cdot (v_x, v_y, v_z) \).
2. Lissafa Samfurin Giciye (\( \vec{r} \times \vec{p} \)):
Ana iya ƙididdige samfurin giciye na vectors guda biyu a cikin daidaitawar Cartesian ta hanyar:
\[
\vec{L} = \vec{r} \times \vec{p} = \left( \begin{array}{c}
y \cdot p_z – z \cdot p_y \\
z \cdot p_x – x \cdot p_z \\
x \cdot p_y – y \cdot p_x \\
\end{array} \right)
\]
3. Kimanta Darajar da Alkiblar Motsin Kungular:
Sakamakon samfurin giciye vector ne mai takamaiman alkibla da girma. Ana iya ƙididdige girman ƙarfin kusurwa ta hanyar ɗaukar girman vector \(\vec{L}\):
\[
|\vec{L}| = \sqrt{(L_x)^2 + (L_y)^2 + (L_z)^2}
\]
Lissafin Ƙarfin Kusurwa a Tsarin Ci gaba
Ga abubuwan da ke da ci gaba da rarraba taro, kamar sandar juyawa ko faifai, mafi yawan matakai sune kamar haka:
1. Ƙayyade Lokacin Rashin Inertia (\( I \)):
Lokacin inertia wani tensor ne wanda ke bayyana yadda ake rarraba nauyin abu dangane da axis na juyawarsa. Wasu misalan lokutan inertia na siffofi daban-daban na abu:
– Dogon sanda \( L \) tare da juyawa a tsakiya: \( I = \frac{1}{12} m L^2 \)
– Faifan da ke da radius \( R \): \( I = \frac{1}{2} m R^2 \)
– Sphere mai ƙarfi tare da radius \( R \): \( I = \frac{2}{5} m R^2 \)
2. Ƙayyade Saurin Kusurwa (\( \omega \)):
Saurin kusurwa shine yadda abu ke juyawa da sauri kuma yawanci ana auna shi da radians a kowace daƙiƙa.
3. Ninkuwa Lokacin Rashin Inertia da Saurin Kusurwa:
Yi amfani da dabarar \( L = I \cdot \omega \) don samun ma'aunin kusurwa na abu.
Misalin matsalolin
Misali na 1: Ƙwayoyin Haɗaka Suna Matsarwa a Layi Madaidaiciya
A ce wani barbashi mai nauyin kilogiram 2 yana tafiya a gudun mita 3/s a cikin alkiblar \( \hat{i} \) kuma yana a matsayi mita 2 daga axis na juyawa a cikin alkiblar \( \hat{j} \).
1. Matsayin vector \( \vec{r} = 2 \hat{j} \)
2. Momentum vector \( \vec{p} = 2 \sau 3 \hat{i} = 6 \hat{i} \)
3. Tsarin samfuri \( \vec{L} = \vec{r} \times \vec{p} \):
\[
\vec{L} = \begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
0 da 2 da 0 \\
6 da 0 da 0 \\
\end{vmatrix} = (0)(0) – (2)(0) \hat{i} – (0)(0) + (6)(0) \hat{j} + (2)(6) – (0)(0) \hat{k}
= (0 \hat{i}, -0 \hat{j}, 12 \hat{k})
= 12 \hat{k}
\]
Don haka, \( \vec{L} = 12 \hat{k} \, \text{kg} \cdot \text{m}^2 / \text{s} \).
Misali na 2: Faifan Juyawa
Faifan da ya yi kama da na diski mai nauyin kilogiram 5 da kuma radius na mita 0.5 yana juyawa tare da saurin kusurwa na radians 10/s.
1. Lokacin rashin kuzari, \( I = \frac{1}{2} m R^2 = \frac{1}{2} \times 5 \times (0.5)^2 = \frac{1}{2} \times 5 \times 0.25 = 0.625 \, \text{kg} \cdot \text{m}^2 \)
2. Saurin kusurwa, \( \omega = 10 \, \text{rad/s} \)
3. Motsin kusurwa, \( L = I \cdot \omega = 0.625 \times 10 = 6.25 \, \text{kg} \cdot \text{m}^2 / \text{s} \)
Amfani da Tsarin Angular
Fahimtar ƙarfin kusurwa yana da amfani iri-iri. Misali:
– Falaki: Nauyin taurarin da ke mutuwa yana sa taurarin da ke kewaye da shi su ci gaba da kasancewa a kusurwarsu, wanda hakan ke da tasiri ga juyawarsu a kusa da tauraron.
– Makamashin Iska: Injinan iska suna amfani da ƙa'idar motsin kusurwa don canza kuzarin motsi na iska zuwa makamashin lantarki.
– Wasanni: 'Yan wasa galibi suna amfani da ƙa'idar motsin kusurwa a cikin motsi daban-daban, kamar juyawa a cikin nutsewa ko jefa mashi.
Kammalawa
Motsin kusurwa wani babban ra'ayi ne mai zurfi kuma mai amfani a fannin kimiyyar lissafi. Ta hanyar fahimtar yadda ake ƙididdige shi don tsarin da ke rarrabe da kuma mai ci gaba, za mu iya samun fahimtar juyawa da daidaiton abubuwa daban-daban. Amfanin wannan ilimin ya wuce ilimin ilimi zuwa aikace-aikacen aiki a rayuwar yau da kullun.
