Tambayoyi Misali Game da Dokar Kare Makamashi
Dokar Kare Makamashi ƙa'ida ce ta asali a fannin kimiyyar lissafi, tana mai cewa makamashi a cikin tsarin da aka rufe ba za a iya ƙirƙira shi ko lalata shi ba, amma ana iya canza shi ne kawai daga wani tsari zuwa wani. Wannan ra'ayi yana da mahimmanci kuma ana amfani da shi akai-akai a fannoni daban-daban na kimiyya da fasaha, gami da makanikai, thermodynamics, da electromagnetism. A cikin wannan labarin, za mu tattauna misalai da yawa na matsaloli da suka shafi Dokar Kare Makamashi, tare da cikakkun bayanai.
Misali Tambaya ta 1: Makamashin Inji a cikin Abu Mai Faɗuwa Kyauta
Tambaya: Ana sauke ƙwallon da nauyinta ya kai kilogiram 0,5 daga tsayin mita 20. Yi watsi da juriyar iska. Menene saurin ƙwallon idan ta isa ƙasa?
Tattaunawa:
Mataki na 1: Gano ƙarfin farko da na ƙarshe.
Ƙarfin farko da ƙwallon ke da shi a tsayin mita 20 shine ƙarfin ƙarfin nauyi (EP), wanda za'a iya ƙididdige shi ta amfani da dabarar:
\[ EP = mgh \]
Ina:
– \( m \) shine nauyin ƙwallon (0,5 kg)
– \( g \) shine hanzarin da aka samu sakamakon nauyi (9,8 m/s²)
– \( h \) shine tsayin (mita 20)
\[ EP_{na farko} = 0,5 \sau 9,8 \sau 20 = 98 \, \text{J} \]
Ƙarfin ƙarshe idan ƙwallon ya isa ƙasa shine kuzarin motsi (EK), wanda za'a iya ƙididdige shi ta amfani da dabarar:
\[ EK = \frac{1}{2} mv^2 \]
Ina:
– \( v \) shine saurin ƙwallon idan ta isa ƙasa
Mataki na 2: Yi amfani da Dokar Kare Makamashi don bayyana cewa ƙarfin farko na makamashin yana daidai da ƙarfin motsi na ƙarshe.
\[ EP_{na farko} = EK_{na ƙarshe} \]
\[ 98 = \frac{1}{2} \sau 0,5 \sau v^2 \]
Mataki na 3: Warware lissafin don nemo \( v \).
\[ 98 = 0,25v^2 \]
\[ v^2 = \ frac{98}{0,25} = 392 \]
\[ v = \sqrt{392} \kimanin 19,8 \, \text{m/s} \]
Don haka, saurin ƙwallon idan ya isa ƙasa yana da kimanin m19,8/s.
Misali Tambaya ta 2: Makamashi a cikin bazara
Tambaya: Maɓuɓɓugar ruwa mai ma'aunin bazara \( k \) = 200 N/m ana matse ta mita 0,1 daga matsayin daidaitonsa. Nawa ne makamashin da ke da ƙarfin roba da aka adana a cikin bazara?
Tattaunawa:
Mataki na 1: Yi amfani da dabarar makamashi mai ƙarfi.
\[ EP_{elastic} = \frac{1}{2} kx^2 \]
Ina:
– \( k \) shine ma'aunin bazara (200 N/m)
– \( x \) shine nisan matsi (0,1 m)
Mataki na 2: Sanya dabi'un da aka sani a cikin dabarar.
\[ EP_{elastic} = \frac{1}{2} \sau 200 \sau (0,1)^2 \]
\[EP_{elastic} = \frac{1}{2} \sau 200 \sau 0,01 \]
\[EP_{elastic} = 1 \, \text{J} \]
Don haka, ƙarfin da ke da ƙarfin roba da aka adana a lokacin bazara shine Joule 1.
Misali Tambaya ta 3: Makamashin Ƙarfin Nauyi da Makamashin Kinetic a cikin Motsin Parabolic
Tambaya: Ana harba wani harsashi mai nauyin kilogiram 2 tare da saurin farko na m 30/s a kusurwar 45° zuwa kwance. Menene kuzarin motsi da yuwuwar na harsashin a mafi girman wurin da yake tafiya?
Tattaunawa:
Mataki na 1: Raba saurin farko zuwa sassa na kwance da kuma a tsaye.
\[ v_{x} = v_0 \cos \theta \]
\[ v_{y} = v_0 \sin \theta \]
Ina:
– \( v_0 \) shine saurin farko (30 m/s)
– \( \theta \) shine kusurwar ƙaddamarwa (45°)
\[ v_{x} = 30 \cos 45° = 30 \times \frac{\sqrt{2}}{2} = 21,21 \, \text{m/s} \]
\[ v_{y} = 30 \sin 45° = 30 \times \frac{\sqrt{2}}{2} = 21,21 \, \text{m/s} \]
Mataki na 2: A mafi girman matsayi, saurin tsaye (v_y) shine 0, amma saurin kwance (v_x) ya kasance daidai.
\[ v_{x \, maki \, mafi girma} = 21,21 \, \text{m/s} \]
Mataki na 3: Lissafa kuzarin motsi na harsashin a mafi girman wurinsa.
\[ EK = \frac{1}{2} mv^2 \]
\[ EK_{maki \, mafi girma} = \frac{1}{2} \sau 2 \sau (21,21)^2 \]
\[ EK_{maki \, mafi girma} = 1 \sau 449,21 = 449,21 \, \text{J} \]
Mataki na 4: Lissafa matsakaicin tsayin da harsashin ya kai.
\[ h = \frac{v_{y}^2}{2g} \]
\[ h = \frac{(21,21)^2}{2 \sau 9,8} \]
\[ h \kimanin 22,9 \, \text{m} \]
Mataki na 5: Lissafa ƙarfin ƙarfin nauyi a mafi girman matsayi.
\[ EP_{maki \, mafi girma} = mgh \]
\[ EP_{maki \, mafi girma} = 2 \sau 9,8 \sau 22,9 \]
\[ EP_{maki \, mafi girma} \kimanin 449,72 \, \text{J} \]
Don haka, kuzarin motsi na harsashi a mafi girman matsayi shine Joules 449,21, kuma ƙarfin ƙarfin nauyi a mafi girman matsayi shine Joules 449,72.
Misali Tambaya ta 4: Makamashin Zafi a Cikin Rashin Daidaito
Tambaya: Akwati mai nauyin kilogiram 10 ana tura shi a kan ƙasa mai kauri na tsawon mita 5 tare da ƙarfin da ke ci gaba da kasancewa na N30. Matsakaicin gogayya tsakanin akwatin da bene shine 0,2. Nawa ne makamashin da aka mayar zuwa makamashin zafi saboda gogayya?
Tattaunawa:
Mataki na 1: Lissafa ƙarfin gogayya.
\[ f_{gogayya} = \mu N \]
Ina:
– \( \mu \) shine ma'aunin gogayya (0,2)
– \( N \) shine ƙarfin da aka saba. Ga saman da aka yi da siminti, \( N = mg \)
\[ f_{gazawa} = 0,2 \sau 10 \sau 9,8 = 19,6 \, \text{N} \]
Mataki na 2: Lissafa aikin da ƙarfin gogayya ya yi.
\[ W = f_{rikice} \times d \]
Ina:
– \( d \) shine nisa (mita 5)
\[ W = 19,6 \sau 5 = 98 \, \text{J} \]
Makamashin da aka canza zuwa makamashin zafi sakamakon gogayya shine Joules 98.
Kammalawa
Dokar Kare Makamashi wata dabara ce mai ƙarfi da ta shafi yanayi da yawa na kimiyyar lissafi. Fahimtar ta tana ba mu damar yin nazari kan abubuwa daban-daban na zahiri, tun daga abubuwan da ke faɗuwa cikin sauƙi da motsin harba har zuwa makamashin da gogayya ke samarwa. Misalan da ke sama suna taimaka mana mu fahimci yadda kuzari zai iya canzawa yayin da yake ci gaba da kasancewa daidai a adadi. Fahimtar wannan ra'ayi yana da matuƙar muhimmanci ga ilimi da aikace-aikacen aiki a fannin injiniyanci da kimiyya.
