Nā Nīnau a me ke Kūkākūkā ʻana no nā Dynamics Rotational
He kumuhana koʻikoʻi ka dynamics rotational i ka physics, e pili ana i ka neʻe ʻana o nā mea e wili ana. Hoʻopuni kēia hanana i nā manaʻo he nui, e like me ka manawa o ka inertia, ka torque, ke kānāwai ʻelua o ka wili ʻana o Newton, ka ikehu kinetic rotational, a me nā mea hou aku. E hōʻike kēia ʻatikala i kekahi mau pilikia a me nā kūkākūkā e pili ana i ka dynamics rotational e wehewehe i ko mākou ʻike i kēia mau manaʻo.
1. Manawa o ka Inertia a me ka Torque
Nīnau:
Ke wili nei kahi apo lahilahi o ke kaumaha \(2\) kg a me ka radius \(0.5\) m a puni kona axis waena. E helu i ka manawa o ka inertia o ke apo, a inā hoʻopili ʻia kahi torque o \(6\) Nm iā ia, he aha ka wikiwiki angular i loaʻa?
Kūkākūkā:
ʻO ke ʻano o ka inertia \(I\) no ke apo lahilahi e pili ana i kona axis waena:
\[ I = MR^2 \]
Me \(M = 2\) kg a me \(R = 0.5\) m, a laila:
\[ I = 2 \times (0.5)^2 \]
\[ I = 2 \times 0.25 \]
\[ I = 0.5 \, \kikokikona{kg} \, \kikokikona{m}^2 \]
Hiki ke helu ʻia ka wikiwiki angular (\(\alpha\)) me ka hoʻohana ʻana i ka torque (\(\tau\)) a me ka moment of inertia (\(I\)):
\[ \tau = I \alpha \]
\[ \alpha = \frac{\tau}{I} \]
Me \(\tau = 6\) Nm a me \(I = 0.5\) kg m²:
\[ \alpha = \frac{6}{0.5} \]
\[ \alpha = 12 \, \text{rad/s}^2 \]
No laila, ʻo ka manawa o ka inertia o ke apo he \(0.5\) kg m² a ʻo ka wikiwiki angular hopena he \(12\) rad/s².
2. Ikehu Kinetic Rotational
Nīnau:
He cylinder paʻa me ka nuipa o \(4\) kg a me ka radius o \(0.3\) m e wili ana me ka wikiwiki angular o \(10\) rad/s. E helu i ka ikehu kinetic rotational o ka cylinder.
Kūkākūkā:
Ua hoʻopili ʻia ka ikehu kinetic rotational (\(K\)) penei:
\[ K = \frac{1}{2} I \omega^2 \]
ʻO \(\omega\) ka wikiwiki angular, a ʻo \(I\) ka manawa o ka inertia. No kahi cylinder paʻa (\(I = \frac{1}{2}MR^2\)):
\[ I = \frac{1}{2} \times 4 \times (0.3)^2 \]
\[ I = 2 \times 0.09 \]
\[ I = 0.18 \, \kikokikona{kg} \, \kikokikona{m}^2 \]
E hoʻololi i nā waiwai o \(I\) a me \(\omega\) i loko o ka hoohalike no ka ikehu kinetic rotational:
\[ K = \frac{1}{2} \times 0.18 \times (10)^2 \]
\[ K = 0.09 \times 100 \]
\[ K = 9 \, \kikokikona{J} \]
No laila, ʻo ka ikehu kinetic rotational o ka cylinder paʻa he \(9\) Joules.
3. Ke Kānāwai ʻElua o ka Hoʻohuli ʻana o Newton
Nīnau:
He radius o ka huila pulley he \(0.4\) m a he manawa inertia o \(0.8\) kg m². Hoʻopili ʻia kahi ikaika \(F\) o \(20\) N ma ke ʻano tangential i ka lihi o ka huila. E helu i ka wikiwiki angular o ka huila.
Kūkākūkā:
ʻO ka mea mua, ke helu nei mākou i ka torque \(\tau\):
\[ \tau = F \times R \]
\[ \tau = 20 \times 0.4 \]
\[ \tau = 8 \, \text{Nm} \]
A laila e hoʻohana i ke kānāwai ʻelua o Newton no ka hoʻololi ʻana:
\[ \tau = I \alpha \]
\[ \alpha = \frac{\tau}{I} \]
Me \(\tau = 8 \, \text{Nm}\) a me \(I = 0.8 \, \text{kg m}^2\):
\[ \alpha = \frac{8}{0.8} \]
\[ \alpha = 10 \, \text{rad/s}^2 \]
No laila, ʻo ka wikiwiki angular o ka huila he \(10\) rad/s².
4. Pilina ma waena o ka Tangent a me ka Unuhi
Nīnau:
ʻO ka pōpō me ke kaumaha o \(1\) kg a me ka radius o \(0.2\) m e ʻōwili me ka ʻole o ka paheʻe ma ka papahele me ka wikiwiki angular o \(5\) rad/s. E helu i ka wikiwiki linear o ke kikowaena o ka nui o ka pōpō.
Kūkākūkā:
ʻO ka pilina ma waena o ka wikiwiki angular (\(\omega\)) a me ka wikiwiki linear (\(v\)) i ka neʻe ʻōwili me ka ʻole o ka paheʻe ʻana penei:
\[ v = \omega R \]
Me \(\omega = 5\) rad/s a me \(R = 0.2\) m:
\[ v = 5 \times 0.2 \]
\[ v = 1 \, \text{m/s} \]
No laila, ʻo ka wikiwiki linear o ke kikowaena o ka nuipa o ka pōpō he \(1\) m/s.
5. Hoʻohuihui ʻia o ka ikehu kinetic Translational a me Rotational
Nīnau:
ʻO kahi cylinder paʻa me ka nui o \(2\) kg a me ka radius o \(0.1\) m e ʻōwili ana i lalo ma kahi mokulele hilig o ke kiʻekiʻe \(3\) m. E helu i ka wikiwiki o ka cylinder ke hiki i ka lalo o ka mokulele hilig.
Kūkākūkā:
Ke ʻōwili ka cylinder me ka paheʻe ʻole, hoʻololi ʻia ka ikehu hiki ke umekaumaha i ikehu kinetic translational a me ka ikehu kinetic rotational. Hiki ke hōʻike ʻia ka ikehu holoʻokoʻa \(E\) penei:
\[ E = mgh = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 \]
Ke noʻonoʻo nei iā \(I\) no kahi cylinder paʻa = \(\frac{1}{2}MR^2\), a me \(\omega = \frac{v}{R}\), lilo ka hoohalike i:
\[ mgh = \frac{1}{2}mv^2 + \frac{1}{2}\left(\frac{1}{2}MR^2\right)\left(\frac{v^2}{R^2}\right) \]
\[ 2 \times 9.8 \times 3 = \frac{1}{2} \times 2 \times v^2 + \frac{1}{2} \left( \frac{1}{2} \times 2 \times v^2 \right) \]
\[ 58.8 = v^2 + \frac{1}{2}v^2 \]
\[ 58.8 = \frac{3}{2}v^2 \]
\[ v^2 = \frac{58.8 \times 2}{3} \]
\[ v^2 = 39.2 \]
\[ v = \sqrt{39.2} \]
\[ v \approx 6.26 \, \text{m/s} \]
No laila, ʻo ka wikiwiki o ka cylinder i kona hiki ʻana i ka lalo o ka mokulele inclined ma kahi o \(6.26\) m/s.
Ka hopena
Ma o kēia mau pilikia, ua ʻimi mākou i kekahi mau manaʻo koʻikoʻi i ka dynamics rotational, me ka manawa o ka inertia, ka torque, ka ikehu kinetic rotational, ke kānāwai ʻelua o ka rotation a Newton, a me ka pilina ma waena o ka neʻe rotational a me ka neʻe translational. ʻO ka hoʻomaopopo maikaʻi ʻana i kēia mau manaʻo e kōkua iā mākou e kālailai a hoʻoponopono i nā pilikia o ke ao maoli e pili ana i ka neʻe rotational.
