Mehlala ea Lipotso tse Buisanang ka Tšebeliso ea Li-Integrals ho Fisiks
Tšebeliso ea lintho tse kopaneng fisiks ke khopolo ea bohlokoa haholo le e pharaletseng. Tšebeliso ea lintho tse kopaneng e lumella litsebi tsa fisiks le baenjiniere ho bala mefuta e fapaneng ea liketsahalo tsa tlhaho tse rarahaneng, ebang li amana le motsamao, matla, matla, kapa likarolo tse ling. Sehlooho sena se tla hlahloba mehlala e 'maloa ea mathata le ho buisana ka ts'ebeliso ea lintho tse kopaneng fisiks.
1. Ho Bala Mosebetsi ka Matla a Fetohang
Soal
Matla a fapaneng ho latela boemo \(x\) a fanoa ke \( F(x) = 3x^2 \). Bala mosebetsi o entsoeng ke matla ana ha ntho e tloha ho \(x = 0\) ho ea ho \(x = 2 \) limithara.
Puisano
Mosebetsi o etsoang ke matla a fetohang ke karolo ea bohlokoa ea matla ho feta sebaka seo. Haeba matla \( F(x) \) e le mosebetsi oa boemo \(x\) a fanoa, re ka hlalosa mosebetsi ka tsela ena:
\[ W = \int_{a}^{b} F(x) \, dx \]
Boemong bona:
\[ F(x) = 3x^2 \]
\[ a = 0 \, \mongolo{mithara} \]
\[ b = 2 \, \mongolo{mithara} \]
Ebe mosebetsi \(W\) ke:
\[ W = \int_{0}^{2} 3x^2 \, dx \]
Re bala karolo ena e kopaneng:
\[
W = 3 \int_{0}^{2} x^2 \, dx
= 3 \left[ \frac{x^3}{3} \right]_{0}^{2}
= 3 \left( \frac{2^3}{3} – \frac{0^3}{3} \right)
= 3 \left( \frac{8}{3} – 0 \right)
= 8 \, \mongolo{Joule}
\]
Kahoo, mosebetsi o entsoeng ke lebotho ke 8 Joules.
2. Ho Bala Bohareng ba Boima ba Thupa e Tšoanang
Soal
Thupa e ts'oanang e bolelele ba \(L\) e fumaneha ho x-axis ho tloha \( x = 0 \) ho isa \( x = L \). Bala boemo ba setsi sa boima ba thupa.
Puisano
Bakeng sa thupa e ts'oanang, boima bo ajoa ka ho lekana ho latela bolelele ba eona. Re ka nahana hore thupa e na le boima bo tsitsitseng ba mola \(\lambda\) (boima ka bolelele ba yuniti).
Bohareng ba boima (\(x_{cm}\)) bo fanoa ke:
\[ x_{cm} = \frac{\int x \, dm}{\int dm} \]
Kaha boima bo aroloa ka ho lekana, re ka hlalosa \(dm = \lambda \, dx\), le moedi o kopaneng ho tloha \(x = 0\) ho isa \(x = L\):
\[
x_{cm} = \frac{\int_{0}^{L} x \lambda \, dx}{\int_{0}^L \lambda \, dx}
\]
Kopanyo hodima \(\lambda\) e dula e le teng mme e ka etsollwa:
\[
x_{cm} = \frac{\int_{0}^{L} x \, dx}{\int_{0}^{L} dx}
= \frac{\left[ \frac{x^2}{2} \right]_{0}^{L}}{ \left[ x \right]_{0}^{L} }
= \frac{\frac{L^2}{2} – 0}{L – 0}
= \frac{L^2 /2}{L}
= \frac{L}{2}
\]
Kahoo, boemo ba setsi sa boima ba molamu bo ho \( \frac{L}{2} \), kapa bohareng ba molamu.
3. Ho Bala Matla a Electrostatic ho latela Molao oa Coulomb
Soal
Litefiso tse peli \(q_1\) le \(q_2\) li fumaneha ho latela x-axis ho \(x = 0\) le \(x = L\) ka ho latellana. Bala matla a motlakase pakeng tsa litefiso tse peli.
Puisano
Molao oa Coulomb o bolela hore matla a pakeng tsa litefiso tse peli tsa lintlha a lekana ka ho toba le sehlahisoa sa litefiso 'me a lekana ka tsela e fapaneng le sekwere sa sebaka se pakeng tsa tsona:
\[ F = k_e \frac{|q_1 q_2|}{r^2} \]
Moo:
– \(k_e\) ke phetolelo e sa fetoheng ya Coulomb \((8.99 \times 10^9 \, \text{N} \cdot \text{m}^2 / \text{C}^2)\)
– \(r\) ke sebaka se pakeng tsa litefiso
Tabeng ena, \(q_1\) le \(q_2\) di ho \(x = 0\) le \(x = L\), ebe sebaka \(r = L\).
Matla a motlakase ke:
\[ F = k_e \frac{|q_1 q_2|}{L^2} \]
Ena ke tharollo e sebelisoang hangata bakeng sa ho bala matla a motlakase pakeng tsa litefiso tse peli tsa lintlha tse behiloeng sebakeng se itseng.
4. Ho Bala Flux ea Magnetic
Soal
Sekotjana sa terata se chitja sa radius \(r\) se beoa tšimong ea makenete e tšoanang \(B\), e otlolohileng ho ea sebakeng sa sekotjana. Bala phallo ea makenete ka sekotjana.
Puisano
Phallo ea makenete (\(\Phi_B\)) e fetang sebakeng \(A\) tšimong ea makenete \(B\) e fanoa ke:
\[ \Phi_B = \int B \cdot dA \]
Kaha matla a makenete \(B\) a lekana ebile a otlolohile ho ya ka sefofane sa sekgoqetsane, karolo e bonolo e ba:
\[ \Phi_B = B \cdot A \]
Moo sebaka sa selikalikoe se nang le radius se leng:
\[ A = \pi r^2 \]
Ebe phallo ea makenete ka har'a selika-likoe ke:
\[ \Phi_B = B \cdot \pi r^2 \]
Kahoo, phallo ea makenete ka har'a selika-likoe ke \( B \pi r^2 \).
Qetello
Tšebeliso ea lintho tse kopaneng fisiks e ke ke ea qojoa ha re tlameha ho bala tlhahisoleseling e amanang le liketsahalo tse rarahaneng tsa tlhaho. Ho tloha ho baleng mosebetsi o etsoang ke matla a feto-fetohang, ho fumana setsi sa boima ba ntho, ho bala matla a motlakase a thehiloeng molaong oa Coulomb, ho isa ho baleng phallo ea makenete ka selika-likoe sa terata tšimong ea makenete, kaofela li itšetlehile ka lintho tse kopaneng ho rarolla mathata. Kutloisiso e felletseng ea hore na lintho tse kopaneng li sebetsa joang maemong a fapaneng a fisiks ha e nolofatse feela tharollo ea mathata empa hape e fana ka temohisiso e tebileng mabapi le mekhoa ea bokahohle boemong ba limolek'hule le boemo ba linaleli.
