Tsarin Hadarin Momentum Impulse
Pengantar
Impulse da momentum muhimman ra'ayoyi ne a fannin kimiyyar lissafi waɗanda suke da mahimmanci don fahimtar abubuwa daban-daban, musamman waɗanda suka shafi karo. A cikin wannan labarin, za mu tattauna ma'anoni, dabaru, da ƙa'idodi da suka shafi impulse, momentum, da karo. Za mu kuma duba misalan lissafi da aikace-aikacen da ake amfani da su a aikace na waɗannan ra'ayoyin.
Ma'anar Ƙarfin Zuciya da Momentum
lokacinta
Momentum (\(p\)) ma'auni ne na adadin motsin da abu yake da shi. Momentum adadi ne na vector wanda ya dogara da taro da saurin abu. A lissafi, ana bayyana momentum kamar haka:
\[p = mv \]
Ina:
– \( p \) shine motsi (kg m/s),
– \( m \) shine nauyin abu (kg),
– \( v \) shine saurin abu (m/s).
Momentum yana nuna yadda yake da wahala a tsayar da abu mai motsi. Girman nauyi ko saurin abu, haka nan ƙarfinsa yake.
Imuls
Impulse (\(I\)) shine canjin motsin da ƙarfin da ke aiki akan wani abu ke samarwa a cikin wani lokaci. Impulse shima adadi ne na vector kuma an bayyana shi azaman:
\[ I = F \Delta t \]
Ina:
– \( I \) shine bugun zuciya (N s ko kg m/s),
– \( F \) shine ƙarfin da ke aiki akan abu (N),
– \( \Delta t \) shine tazara tsakanin lokacin da ƙarfin ke aiki (s).
Impulse yayi daidai da canjin motsin abu:
\[ I = \Delta p = p_f – p_i \]
Ina:
– \( \Delta p \) shine canjin motsi (kg m/s),
– \( p_f \) shine ƙarfin ƙarshe (kg m/s),
– \( p_i \) shine farkon motsi (kg m/s).
Karo
Haɗari wata mu'amala ce da abubuwa biyu ko fiye ke musayar ƙarfin motsi. Ana iya rarraba karo zuwa manyan nau'i biyu: karo mai laushi da karo mai laushi.
Karo Mai Ragewa
A cikin karo na roba, jimlar kuzarin motsi na tsarin kafin da bayan karo ya kasance iri ɗaya. Wannan yana nufin cewa babu wani kuzarin motsi da ke ɓacewa kamar zafi, sauti, ko nakasa ta dindindin. Dokokin kiyaye motsi da kiyaye kuzarin motsi suna aiki a cikin karo na roba.
Karo Mai Rashin Nasaba
A cikin karo na inelastic, wasu daga cikin kuzarin motsi na tsarin suna ɓacewa azaman wani makamashi (misali, zafi, sauti, ko nakasawar abubuwan). Duk da cewa dokar kiyaye ƙarfin motsi har yanzu tana aiki, jimlar kuzarin motsi ba a kiyaye shi ba.
Muhimman Dabaru
Kiyaye Momentum
Dokar kiyaye ƙarfin motsi ta bayyana cewa jimillar ƙarfin tsarin kafin karo ya yi daidai da jimillar ƙarfin tsarin bayan karo, matuƙar babu wani ƙarfin waje da ke aiki akan tsarin:
\[ m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f} \]
Ina:
– \( m_1 \) da \( m_2 \) su ne tarin abu 1 da abu 2 (kg),
– \( v_{1i} \) da \( v_{2i} \) sune saurin farko na abu 1 da abu 2 (m/s),
– \( v_{1f} \) da \( v_{2f} \) su ne saurin ƙarshe na abu na 1 da abu na 2 (m/s).
Kiyaye Makamashin Kinetic (don Hadarin Elastic)
Ga karo mai laushi, jimlar kuzarin motsi na tsarin kafin da kuma bayan karo ya kasance iri ɗaya:
\[ \frac{1}{2} m_1 v_{1i}^2 + \frac{1}{2} m_2 v_{2i}^2 = \frac{1}{2} m_1 v_{1f}^2 + \frac{1}{2} m_2 v_{2f}^2 \]
Misalin Lissafi
Bari mu duba wasu misalan lissafin don fahimtar yadda waɗannan dabarun suke aiki a cikin yanayi na gaske.
Misali na 1: Karo mara ƙarfi
A ce motoci biyu, kowannensu yana da nauyin kilogiram 1000, suna tafiya zuwa ga juna a gudun mita 10/s da mita 15/s, bi da bi. Bayan karo, motocin biyu suna tafiya tare da saurin ƙarshe iri ɗaya. Muna son mu tantance wannan saurin ƙarshe.
1. Jimlar ƙarfin farko na tsarin:
\[ p_{jimlar\_initial} = m_1 v_{1i} + m_2 v_{2i} \]
\[ p_{jimlar\_initial} = 1000 \sau 10 + 1000 \sau (-15) \]
\[ p_{jimlar\_initial} = 10000 – 15000 \]
\[ p_{jimlar\_initial} = -5000 \, \text{kg m/s} \]
2. Bayan karo, motocin biyu suna tafiya tare don jimlar nauyin ya kasance \(m_1 + m_2 \), kuma saurin ƙarshe shine \(v_f \):
\[ p_{jimlar\_karshe} = (m_1 + m_2) v_f \]
\[ -5000 = (1000 + 1000) v_f \]
\[ -5000 = 2000 v_f \]
\[ v_f = \frac{-5000}{2000} \]
\[ v_f = -2.5 \, \text{m/s} \]
Gudu na ƙarshe na motocin biyu bayan karo shine -2.5 m/s, ma'ana suna tafiya tare a hanya ɗaya a gudun 2.5 m/s a farkon alkiblar motar ta biyu.
Misali na 2: Karo Mai Ragewa
A ce ƙwallon da nauyinta ya kai kilogiram 2 tana tafiya zuwa dama a gudun mita 4/s ta yi karo da wata ƙwallon da nauyinta ya kai kilogiram 3 tana tafiya zuwa hagu a gudun mita 2/s. Muna son mu tantance saurin ƙarshe na ƙwallon biyu bayan karo.
1. Jimlar ƙarfin farko na tsarin:
\[ p_{jimlar\_initial} = m_1 v_{1i} + m_2 v_{2i} \]
\[ p_{jimlar\_initial} = 2 \sau 4 + 3 \sau (-2) \]
\[ p_{jimlar\_initial} = 8 – 6 \]
\[ p_{jimlar\_initial} = 2 \, \text{kg m/s} \]
2. Jimillar kuzarin motsi na tsarin kafin karo:
\[ KE_{total\_initial} = \ frac{1}{2} m_1 v_{1i}^2 + \frac{1}{2} m_2 v_{2i}^2 \]
\[ KE_{total\_initial} = \frac{1}{2} \sau 2 \sau 4^2 + \frac{1}{2} \sau 3 \sau 2^2 \]
\[ KE_{total\_farko} = 16 + 6 \]
\[ KE_{total\_initial} = 22 \, \rubutu{J} \]
3. Bayan karo, muna buƙatar warware daidaiton kiyayewa na ƙarfin motsi da kuzarin motsi a lokaci guda don nemo saurin ƙarshe \(v_{1f}\) da \(v_{2f}\).
\[
\begin{cases}
m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f} \\
\frac{1}{2} m_1 v_{1i}^2 + \frac{1}{2} m_2 v_{2i}^2 = \frac{1}{2} m_1 v_{1f}^2 + \frac{1}{2} m_2 v_{2f}^2
\ƙarshen
\]
Ta hanyar maye gurbin da lissafi, za mu iya samun saurin ƙarshe na ƙwallo biyu. Sakamakon ƙarshe shine:
\[ v_{1f} \approx -2.2 \, \text{m/s} \]
\[ v_{2f} \approx 3.2 \, \text{m/s} \]
Don haka, bayan karo mai laushi, ƙwallon farko ta motsa zuwa hagu da kimanin m2.2/s, ƙwallon ta biyu kuma ta motsa zuwa dama da kimanin m3.2/s.
Aikace-aikace Masu Amfani
1. Motoci da Tsaro
Manufofin motsin rai da ƙarfin motsi suna da matuƙar muhimmanci a tsarin tsaron motoci. An tsara jakunkunan iska da wuraren da ke da rauni don tsawaita lokacin tasirin, rage ƙarfin da ke kan masu ababen hawa, da kuma rage raunuka.
2. Wasanni
A wasanni kamar ƙwallon ƙafa, dambe, da hockey, fahimtar motsin rai da ƙarfin hali yana taimaka wa 'yan wasa su inganta ayyukansu. Misali, a cikin dambe, bugun da ya dace ya ƙunshi haɓaka canjin ƙarfin hali cikin ɗan gajeren lokaci.
3. Injiniyan Gine-gine da Zane
Injiniyoyin suna amfani da ƙa'idodin motsin rai da ƙarfin aiki don tsara gine-gine waɗanda za su iya jure wa lodi mai ƙarfi, kamar gadoji da manyan gine-gine, da kuma tabbatar da kwanciyar hankali da amincin gine-gine yayin tasirin ko girgiza.
