Sida Loo Xisaabiyo Dhaqdhaqaaqa Angular
Mowjadda Angular waa fikrad muhiim ah oo ku jirta fiisikiska, gaar ahaan farsamada casriga ah iyo kuwa quantum-ka. Maqaalkan, waxaan si faahfaahsan uga hadli doonnaa sida loo xisaabiyo mowjadda angular, hababka kala duwan ee la heli karo, iyo adeegsigeeda nolol maalmeedka. Fahmidda fikraddan waxay faa'iido u leedahay ardayda fiisikiska iyo xirfadlayaasha, laakiin sidoo kale qof kasta oo xiiseynaya sida dabeecaddu u shaqeyso heer aasaasi ah.
Pendahuluan
Mowjadda xagasha waa tiro vector ah oo qeexaysa wareegga shay ku wareegsan barta. Sida mowjadda toosan ay ula xiriirto dhaqdhaqaaqa toosan, mowjadda xagasha waxay maamushaa sida shay u wareego. Qaacidada aasaasiga ah ee mowjadda xagasha (\(L\)) waa natiijada daqiiqadda inertia (\(I\)) iyo xawaaraha xagasha (\(\omega\)):
\[ L = I \cdot \omega \]
Si kastaba ha ahaatee, haddii aan tixgelinno kiiska walax ku wareegaysa barta, qaacidada la isticmaalay waa:
\[ L = r \times p \]
Halkee:
– \( r \) waa vector-ka booska ee walaxda marka loo eego bartamaha wareegga.
– \( p \) waa xawaaraha toosan ee walaxda (\( p = m \cdot v \) halkaas oo \( m \) uu yahay cufka walaxda iyo \( v \) uu yahay xawaaraha toosan).
Calaamadda "\(\times\)" waxay matalaysaa wax soo saarka iskutallaabta ee vectors-ka, taasoo macnaheedu yahay in dhaqdhaqaaqa xagasha uu had iyo jeer ku toosan yahay diyaaradda ay sameeyeen vector-ka booska \( r \) iyo vector-ka dhaqdhaqaaqa \( p \).
Xisaabinta Dhaqdhaqaaqa Angular ee Nidaamyada Kala Duwan
Ka soo qaad inaan haysanno walax leh cuf \( m \) oo ku socota xawaare \( v \) meel fog \( r \) oo ka fog bartamaha wareegga. Tallaabooyinka lagu xisaabinayo dhaqdhaqaaqa xagasha waa sidan soo socota:
1. Go'aami Meesha uu ku yaal Vektor (\( r \)) iyo Momentum Vektor (\( p \)):
Hubi in dhammaan vektors-ka laga cabbiro bartamaha wareegga. Ka soo qaad in walaxdu ay ku jirto booska \( (x, y, z) \) oo ay ku socoto xawaare \( (v_x, v_y, v_z) \). Kadib, vektor-ka booska waa \( \vec{r} = (x, y, z) \), vektor-ka dhaqdhaqaaqana waa \( \vec{p} = m \cdot (v_x, v_y, v_z) \).
2. Xisaabi Badeecada Iskutallaabta ah (\( \vec{r} \times \vec{p} \)):
Wax soo saarka isdhaafka ah ee laba vectors ee isku-duwayaasha Cartesian waxaa lagu xisaabin karaa:
\[
\vec{L} = \vec{r} \times \vec{p} = \left( \begin{array}{c}
y \cdot p_z – z \cdot p_y \\
z \cdot p_x – x \cdot p_z \\
x \cdot p_y – y \cdot p_x \\
\end{array} \right)
\]
3. Qiimaynta Qiimaha iyo Jihada Dhaqdhaqaaqa Angular:
Natiijada ka soo baxda badeecada isdhaafka ah waa vektor leh jiho iyo baaxad gaar ah. Cabbirka dhaqdhaqaaqa xagasha waxaa lagu xisaabin karaa iyadoo la qaadanayo baaxadda vektorka \(\vec{L}\):
\[
|\vec{L}| = \sqrt{(L_x)^2 + (L_y)^2 + (L_z)^2}
\]
Xisaabinta Dhaqdhaqaaqa Angular ee Nidaamyada Joogtada ah
Walxaha leh qaybin cuf oo joogto ah, sida ul ama disk wareegaysa, tallaabooyinka guud waa sidan soo socota:
1. Go'aami Waqtiga Inertia (\( I \)):
Daqiiqadda inertia waa tensor qeexaya sida cufka shaygu u qaybsamo marka loo eego dhidibka wareegga. Tusaalooyin qaar oo ah daqiiqado inertia ah oo loogu talagalay qaabab kala duwan oo shay ah:
– Ul dheer \( L \) oo leh wareeg dhexda ah: \( I = \frac{1}{12} m L^2 \)
– Saxan leh gacanka \( R \): \( I = \frac{1}{2} m R^2 \)
– Goobo adag oo leh gacan \( R \): \( I = \frac{2}{5} m R^2 \)
2. Go'aami Xawaaraha Xaglaha (\( \omega \)):
Xawaaraha xaglaha waa sida ugu dhakhsaha badan ee shay u wareego waxaana badanaa lagu cabbiraa radiyaan ilbiriqsi kasta.
3. Ku dhufo Daqiiqada Inertia Xawaaraha Xaglaha:
Adeegso qaacidada \( L = I \cdot \omega \) si aad u hesho xawaaraha xagasha ee shayga.
Tusaalaha dhibaatooyinka
Tusaale 1: Walxaha Ku Socda Xariiq Toosan
Ka soo qaad in walax miisaankeedu yahay 2 kg ay ku socoto xawaare dhan 3 m/s jihada \( \hat{i} \) oo ay ku jirto boos 2 mitir u jira dhidibka wareegga jihada \( \hat{j} \).
1. Vektor-ka booska \( \vec{r} = 2 \hat{j} \)
2. Vektor-ka dhaqdhaqaaqa \( \vec{p} = 2 \jeer 3 \hat{i} = 6 \hat{i} \)
3. Isdhaafsiga badeecada \( \vec{L} = \vec{r} \times \vec{p} \):
\[
\vec{L} = \begin{vmatrix}
\koofiyad{i} & \koofiyad{j} & \koofiyad{k} \\
0 & 2 & 0 \\
6 & 0 & 0 \\
\end{vmatrix} = (0)(0) – (2)(0) \hat{i} – (0)(0) + (6)(0) \hat{j} + (2)(6) – (0)(0) \hat{k}
= (0 \hat{i}, -0 \hat{j}, 12 \hat{k})
= 12 \koofiyad{k}
\]
Markaa, \( \vec{L} = 12 \hat{k} \, \text{kg} \cdot \text{m}^2 / \text{s} \).
Tusaale 2: Saxanka Wareegaya
Saxan isku mid ah oo miisaankiisu yahay 5 kg iyo gacan 0.5 mitir ah ayaa wareegaya iyadoo xawaare xagal ah uu yahay 10 radian/s.
1. Daqiiqadda firfircoonida, \( I = \frac{1}{2} m R^2 = \frac{1}{2} \times 5 \times (0.5)^2 = \frac{1}{2} \times 5 \times 0.25 = 0.625 \, \text{kg} \cdot \text{m}^2 \)
2. Xawaaraha xagasha, \( \omega = 10 \, \text{rad/s} \)
3. Dhaqdhaqaaqa xagasha, \( L = I \cdot \omega = 0.625 \times 10 = 6.25 \, \text{kg} \cdot \text{m}^2 / \text{s} \)
Adeegsiga Dhaqdhaqaaqa Angular
Fahmidda xawaaraha xagasha waxay leedahay adeegyo kala duwan oo wax ku ool ah. Tusaale ahaan:
– Fiisigiska Xiddigiska: Cufisjiidadka xiddig dhimanaya ayaa sababa in meerayaasha ku hareeraysan ay sii haystaan dhaqdhaqaaqooda xagalaysan, taas oo saameyn ku leh wareegga ay ku wareegayaan xiddigta.
– Tamarta Dabaysha: Marawaxadaha dabayshu waxay adeegsadaan mabda'a dhaqdhaqaaqa xagal-xagal si ay tamarta dhaqdhaqaaqa dabaysha ugu beddelaan tamar koronto.
– Ciyaaraha: Ciyaartoydu waxay inta badan adeegsadaan mabda'a dhaqdhaqaaqa xagasha dhaqdhaqaaqyo kala duwan, sida wareegga quusitaanka ama tuurista waranka.
Gabagabo
Dhaqdhaqaaqa xagasha waa fikrad qoto dheer oo khuseysa fiisikiska. Marka aan fahamno sida loo xisaabiyo nidaamyada kala duwan iyo kuwa joogtada ah, waxaan si cad u fahmi karnaa wareegga iyo dheelitirka walxaha kala duwan. Faa'iidooyinka aqoontani waxay ka baxsan yihiin tacliinta ilaa codsiyada wax ku oolka ah ee nolol maalmeedka.
