Tambayoyi da Tattaunawa game da Tsarin Juyawa
Tsarin juyawa muhimmin batu ne a fannin kimiyyar lissafi, wanda ke magana game da motsin abubuwa masu juyawa. Wannan lamari ya ƙunshi ra'ayoyi da yawa, kamar lokacin inertia, ƙarfin juyi, dokar juyawa ta biyu ta Newton, kuzarin motsi na juyawa, da ƙari. Wannan labarin zai gabatar da matsaloli da tattaunawa da dama kan tsarin juyawa don fayyace fahimtarmu game da waɗannan ra'ayoyi.
1. Lokacin Inertia da Ƙarfin Juyawa
Tambaya:
Zoben siriri mai nauyin kilogiram 2 da radius 0.5 m yana juyawa a tsakiyar axis ɗinsa. Lissafa lokacin inertia na zoben, kuma idan aka yi amfani da ƙarfin juyi na ‑6 Nm a kansa, menene sakamakon haɓaka kusurwa?
Tattaunawa:
Lokacin inertia \(I\) ga siririn zobe a tsakiyar tsakiyarsa shine:
\[ Ni = MR^2 \]
Da \(M = 2\) kg da \(R = 0.5\) m, to:
\[ I = sau 2 (0.5)^2 \]
\[ I = sau 2 0.25 \]
\[ I = 0.5 \, \text{kg} \, \text{m}^2 \]
Ana iya ƙididdige hanzarin kusurwa (\(\alpha\)) ta amfani da ƙarfin juyi (\(\tau\)) da lokacin inertia (\(I\)):
\[ \tau = I \alpha \]
\[ \alpha = \frac{\tau}{I} \]
Tare da \(\tau = 6\) Nm da \(I = 0.5\) kg m²:
\[ \alpha = \frac{6}{0.5} \]
\[ \alpha = 12 \, \text{rad/s}^2 \]
Don haka, lokacin inertia na zoben shine 0.5 kg m² kuma saurin kusurwar da ke haifarwa shine 12 rad/s².
2. Makamashin Kinetic na Juyawa
Tambaya:
Silinda mai ƙarfi mai nauyin kilogiram 4 da kuma radius na m (0.3) yana juyawa tare da saurin kusurwa na rad/s. Lissafa kuzarin motsi na silinda.
Tattaunawa:
An tsara makamashin motsi na juyawa (\(K\)) kamar haka:
\[ K = \frac{1}{2} I \omega^2 \]
Inda \(\omega\) shine saurin kusurwa, kuma \(I\) shine lokacin inertia. Ga silinda mai ƙarfi (\(I = \frac{1}{2}MR^2\)):
\[ I = \frac{1}{2} \sau 4 \sau (0.3)^2 \]
\[ I = sau 2 0.09 \]
\[ I = 0.18 \, \text{kg} \, \text{m}^2 \]
Sauya ƙimar \(I\) da \(\omega\) zuwa cikin lissafin kuzarin motsi na juyawa:
\[ K = \frac{1}{2} \sau 0.18 \sau (10)^2 \]
\[ K = 0.09 \sau 100 \]
\[ K = 9 \, \rubutu{J} \]
Don haka, kuzarin motsi na juyawa na silinda mai ƙarfi shine Joules.
3. Dokar Juyawa ta Biyu ta Newton
Tambaya:
Tayar kura tana da radius na m (0.4) da kuma lokacin inertia na m (0.8) kg (m²). Ana amfani da ƙarfin N (F) na N a gefen tayoyin. A ƙididdige hanzarin kusurwa na tayoyin.
Tattaunawa:
Da farko, muna ƙididdige ƙarfin juyi \(\tau\):
\[ \tau = F \times R \]
\[ \tau = 20 \sau 0.4 \]
\[ \tau = 8 \, \text{Nm} \]
Sai a yi amfani da dokar Newton ta biyu don juyawa:
\[ \tau = I \alpha \]
\[ \alpha = \frac{\tau}{I} \]
Tare da \(\tau = 8 \, \text{Nm}\) da \(I = 0.8 \, \text{kg m}^2\):
\[ \alpha = \frac{8}{0.8} \]
\[ \alpha = 10 \, \text{rad/s}^2 \]
Don haka, hanzarin kusurwa na tayoyin shine 10 rad/s².
4. Alaƙa tsakanin Tangent da Fassara
Tambaya:
Kwallo mai nauyin kilogiram 1 da kuma radius na mita 0.2 tana birgima ba tare da zamewa a ƙasa mai saurin kusurwa na rad/s ba. Lissafa saurin layi na tsakiyar nauyin ƙwallon.
Tattaunawa:
Alaƙar da ke tsakanin saurin kusurwa (\(\omega\)) da saurin layi (\(v\)) a cikin motsi ba tare da zamewa ba ita ce:
\[ v = \omega R \]
Tare da \(\omega = 5\) rad/s da \(R = 0.2\) m:
\[ v = 5 \sau 0.2 \]
\[ v = 1 \, \rubutu{m/s} \]
Don haka, saurin layi na tsakiyar nauyin ƙwallon shine \(1\) m/s.
5. Haɗuwar Makamashin Juyawa na Fassara da Juyawa
Tambaya:
Silinda mai ƙarfi mai nauyin kilogiram 2 da radius na mita 0.1 yana birgima ƙasa da tsayin mita 3. Lissafa saurin silinda lokacin da ya kai ƙasan jirgin da aka karkata.
Tattaunawa:
Idan silinda ta yi birgima ba tare da zamewa ba, makamashin ƙarfin nauyi yana canzawa zuwa makamashin motsi na fassara da makamashin motsi na juyawa. Jimlar kuzarin \(E\) za a iya bayyana shi kamar haka:
\[ E = mgh = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 \]
Idan aka yi la'akari da \(I\) don silinda mai ƙarfi = \(\frac{1}{2}MR^2\), da \(\omega = \frac{v}{R}\), lissafin ya zama:
\[ mgh = \frac{1}{2}mv^2 + \frac{1}{2}\left(\frac{1}{2}MR^2\right)\left(\frac{v^2}{R^2}\right) \]
\[ 2 \sau 9.8 \sau 3 = \frac{1}{2} \sau 2 \sau v^2 + \frac{1}{2} \left( \frac{1}{2} \sau 2 \sau v^2 \right) \]
\[58.8 = v^2 + \frac{1}{2}v^2 \]
\[ 58.8 = \frac{3}{2}v^2 \]
\[ v^2 = \frac{58.8 \sau 2}{3} \]
\[ v^2 = 39.2 \]
\[ v = \sqrt{39.2} \]
\[ v \kimanin 6.26 \, \text{m/s} \]
Don haka, saurin silinda idan ya kai ƙasan jirgin da aka karkata yana kusan m(6.26\) m/s.
Kammalawa
Ta hanyar waɗannan matsalolin, mun bincika muhimman ra'ayoyi da dama a cikin yanayin juyawa, ciki har da lokacin inertia, ƙarfin juyi, kuzarin motsi na juyawa, dokar juyawa ta biyu ta Newton, da kuma alaƙar da ke tsakanin motsi na juyawa da na fassara. Fahimtar waɗannan ra'ayoyi zai taimaka mana mu bincika da kuma magance matsalolin da suka shafi motsi na juyawa.
