Dokar Newton ta Biyu tana ɗaya daga cikin muhimman ra'ayoyi a fannin kimiyyar lissafi waɗanda ke kula da alaƙar da ke tsakanin ƙarfi, taro, da hanzari. Wannan dokar ta bayyana cewa hanzarta abu yana daidai da ƙarfin da ke aiki a kansa kuma yana daidai da nauyinsa. A lissafi, an bayyana Dokar Newton ta Biyu kamar haka:
\[ F = ma \]
Ina:
– \( F \) shine ƙarfin da ke aiki akan abu (a cikin Newtons, N).
– \( m \) shine nauyin abu (a cikin kilogiram, kg).
– \( a \) shine hanzarin abu (a cikin mita a kowace daƙiƙa murabba'i, \( m/s^2 \)).
A cikin wannan labarin, za mu tattauna wasu misalai na Dokar Biyu ta Newton don fahimtar amfaninta a cikin yanayi daban-daban.
Misali Tambaya ta 1: Tilasta wa Mota Mai Saurin Hawan Mota
Tambaya:
Mota mai nauyin kilogiram 1000 tana sauri daga hutawa zuwa gudun mita 20/s cikin daƙiƙa 5. Lissafa ƙarfin da ake buƙata don cimma wannan hanzarin.
Mafita:
Da farko, muna buƙatar ƙididdige saurin motar. Ana iya ƙididdige saurin (\( a \)) ta amfani da dabarar:
\[a = \frac{\Delta v}{\Delta t} \]
Ina:
– \(\Delta v\) shine canjin saurin.
– \(\Delta t\) shine canjin lokaci.
Maye gurbin dabi'un da aka sani:
\[ a = \frac{20 \, \text{m/s} – 0 \, \text{m/s}}{5 \, \text{seconds}} \]
\[ a = \frac{20 \, \text{m/s}}{5 \, \text{seconds}} \]
\[a = 4 \, \rubutu{m/s}^2 \]
Yanzu, za mu iya ƙididdige ƙarfin da ake buƙata ta amfani da Dokar Biyu ta Newton:
\[ F = ma \]
\[ F = (1000 \, \text{kg})(4 \, \text{m/s}^2) \]
\[ F = 4000 \, \rubutu{N} \]
Don haka, ƙarfin da ake buƙata don hanzarta motar shine N4000.
Misali Tambaya ta 2: Ƙarfin da ke Kan Akwati
Tambaya:
Ana tura akwati mai nauyin kilogiram 50 a kan wani wuri mai tsauri mai ƙarfin N300. Idan ƙarfin gogayya tsakanin akwatin da saman shine N100, ƙididdige hanzarin akwatin.
Mafita:
Da farko, muna ƙididdige ƙarfin da ke aiki a kan akwatin. Ƙarfin da ke aiki a kan akwatin (\( F_{\text{net}} \)) shine jimillar ƙarfin da ke aiki a kan wani abu bayan an yi la'akari da dukkan ƙarfin da ke aiki a kansa, gami da gogayya.
\[ F_{\text{net}} = F_{\text{push}} – F_{\text{comaction}} \]
\[ F_{\text{net}} = 300 \, \text{N} – 100 \, \text{N} \]
\[ F_{\text{net}} = 200 \, \text{N} \]
Yanzu, za mu iya ƙididdige hanzarin akwatin ta amfani da Dokar Biyu ta Newton:
\[F_{\text{net}} = ma \]
\[ 200 \, \text{N} = (50 \, \text{kg})a \]
\[a = \frac{200 \, \text{N}}{50 \, \text{kg}} \]
\[a = 4 \, \rubutu{m/s}^2 \]
Saboda haka, saurin akwatin shine 4 m/s².
Misali Tambaya ta 3: Lissafin Ƙarfin da ake Bukata don Ɗaga Kaya
Tambaya:
Gilashi yana ɗaga kaya mai nauyin kilogiram 200 sama tare da hanzarin mita 1,5/s². Lissafa ƙarfin da gilashin ke buƙata don ɗaga kaya.
Mafita:
Da farko, muna buƙatar ƙididdige ƙarfin nauyi da ke aiki akan nauyin. Ana iya ƙididdige ƙarfin nauyi (\( F_g \)) ta amfani da dabarar:
\[ F_g = mg \]
Ina:
– \( g \) shine hanzarin da aka samu sakamakon nauyi (\( 9,8 \, \text{m/s}^2 \)).
Maye gurbin dabi'un da aka sani:
\[ F_g = (200 \, \text{kg})(9,8 \, \text{m/s}^2) \]
\[ F_g = 1960 \, \rubutu{N} \]
Yanzu, muna ƙididdige jimlar ƙarfin (\( F \)) da crane ke buƙata don ɗaga nauyin la'akari da ƙarin hanzarin:
\[ F = ma + F_g \]
\[ F = (200 \, \text{kg})(1,5 \, \text{m/s}^2) + 1960 \, \text{N} \]
\[ F = 300 \, \rubutu{N} + 1960 \, \rubutu{N} \]
\[ F = 2260 \, \rubutu{N} \]
Don haka, ƙarfin da crane ke buƙata don ɗaga nauyin shine 2260 N.
Misali Matsala ta 4: Ƙarfi a Tsarin Abubuwa Biyu da Igiya ta Haɗa
Tambaya:
Ana haɗa abubuwa biyu masu nauyin kilogiram 10 da kilogiram 20 bi da bi ta hanyar igiya mai sauƙi kuma an rataye su daga pulley. A ƙididdige saurin tsarin da kuma matsin lamba a cikin igiyar lokacin da aka saki tsarin daga hutawa.
Mafita:
Da farko, bari mu bayyana ƙarfin da ke aiki akan abubuwa biyu. Bari mu kira taro \( m_1 = 10 \, \text{kg} \) da taro \( m_2 = 20 \, \text{kg} \). Ƙarfin nauyi da ke aiki akan abubuwa biyu shine:
\[ F_{g1} = m_1 g = (10 \, \text{kg})(9,8 \, \text{m/s}^2) = 98 \, \text{N} \]
\[ F_{g2} = m_2 g = (20 \, \text{kg})(9,8 \, \text{m/s}^2) = 196 \, \text{N} \]
Tunda tsarin ya fito daga hutu, ana iya ƙididdige saurin tsarin ta amfani da Dokar Biyu ta Newton. Jimlar hanzarin (\( a \)) na tsarin shine:
\[ (m_1 + m_2)a = F_{g2} – F_{g1} \]
\[ (10 \, \text{kg} + 20 \, \text{kg})a = 196 \, \text{N} – 98 \, \text{N} \]
\[ 30 \, \text{kg} \cdot a = 98 \, \text{N} \]
\[a = \frac{98 \, \text{N}}{30 \, \text{kg}} \]
\[a = 3,27 \, \rubutu{m/s}^2 \]
Yanzu, mun ƙididdige tashin hankali a cikin zaren (\( T \)). Ana iya ƙididdige tashin hankali a cikin zaren ta amfani da Dokar Biyu ta Newton akan ɗaya daga cikin taro, misali \( m_1 \):
\[ T – m_1 g = m_1 a \]
\[ T – 98 \, \text{N} = (10 \, \text{kg})(3,27 \, \text{m/s}^2) \]
\[ T – 98 \, \text{N} = 32,7 \, \text{N} \]
\[ T = 32,7 \, \rubutu{N} + 98 \, \rubutu{N} \]
\[ T = 130,7 \, \rubutu{N} \]
Don haka, saurin tsarin shine 3,27 m/s² kuma matsin lamba a cikin igiyar shine 130,7 N.
Kammalawa
Ta hanyar misalai daban-daban na Dokar Biyu ta Newton, mun koyi yadda ake amfani da wannan ƙa'ida don ƙididdige ƙarfi, hanzari, da damuwa a cikin yanayi daban-daban. Dokar Biyu ta Newton ba wai kawai tana da mahimmanci a fannin kimiyyar lissafi ba, har ma tana da aikace-aikace da yawa a rayuwar yau da kullun da fasaha. Ta hanyar fahimtar da kuma aiwatar da Dokar Biyu ta Newton, za mu iya magance matsalolin injiniya daban-daban cikin inganci da daidaito.
