ʻAtikala e pili ana i ka Hoʻohui ʻana o nā Vectors
1. Nā nui o ka vector a me ka scalar
Ma waho aʻe o nā nui kumu a me nā nui i loaʻa, hiki ke hoʻokaʻawale ʻia nā nui kino i ʻelua ʻano ʻē aʻe, ʻo ia hoʻi nā nui scalar a me nā nui vector. ʻO nā nui e like me ka nuipa, ka mamao, ka manawa a me ka leo, he mau nui scalar, nā nui i loaʻa wale ka nui akā ʻaʻohe kuhikuhi. ʻOiai ʻo nā nui e like me ka neʻe ʻana, ka wikiwiki, ka wikiwiki, a me ka ikaika he mau nui vector, nā nui i loaʻa ka nui a loaʻa pū kekahi kuhikuhi.
a. ʻOkoʻa ma waena o ka nui calar a me ka nui vector
Inā ʻōlelo ʻoe he 400 grams ke kaumaha o ka pōpō, ua lawa kēia ʻōlelo nou e ʻike ai i ke kaumaha o ka pōpō. ʻAʻole pono ʻoe i ke kuhikuhi e ʻike ai i ke kaumaha o ka pōpō. Pēlā nō me ka manawa, ka mahana, ka nui, ka nui, a pēlā aku. Aia kekahi mau nui kino i hiki ʻole ke hōʻike ʻia ma ka nui wale nō. Inā ʻōlelo ʻoe e neʻe ana ke keiki a hiki i 100 mika, a laila ʻaʻole lawa kēia ʻōlelo. E nīnau paha ʻoe, ma hea ʻo ia i neʻe ai? He ʻākau, he hema, he hikina, a he komohana paha? Pēlā nō, inā ʻōlelo ʻoe e paʻi ʻoe i ka papaʻaina me ka ikaika o 200 N.
Ma hea ʻoe e kalaiwa ai? ʻĀ, ua kapa ʻia kēlā mau nui he mau nui vector, e pono ai ka wehewehe ʻana i ka nui a me ke kuhikuhi. ʻO nā hiʻohiʻona o nā nui vector ka neʻe ʻana, ka wikiwiki, ka impulse, ka momentum , a pēlā aku. Hiki iā ʻoe ke hoʻomaopopo pono iā ia ke aʻo ʻana i nā kumuhana e pili ana i kēlā mau nui.
b. E kaha kiʻi i kahi vector
Hōʻike ʻia ke vector e kahi pua. Ua kaha mau ʻia ka pua i mea e kuhikuhi ai i ke kuhikuhi o ka vector. Ua wehewehe ʻia ka lōʻihi o ka pua e like me ka nui o ka vector. No ka laʻana, ma ke kiʻi o kahi vector o ka ikaika (F) me ka nui o 2 N nona ke kuhikuhi e kū ana i ka hikina ʻākau a i ʻole 45 o e pili ana i ka axis-x.
c. Nā lula no ke kākau ʻana i nā nui o ka vector
I ke kākau ʻana i kahi vector, inā ʻoe e hoʻohana i ke kākau lima ʻana, ua kākau ʻia ka hōʻailona o kahi vector ma ka italics me ka hoʻohana ʻana i nā leka nui, a ma luna pono e hoʻohui ʻia me kahi pua. No nā puke i paʻi ʻia, ua kākau ʻia nā hōʻailona vector ma nā leka nui me ka wiwo ʻole, no ka laʻana, F. No ka nui o ka vector, inā mākou e hoʻohana i ke kākau lima ʻana a laila ua kākau ʻia ka nui o kahi vector, no ka laʻana, |F|. No nā puke i paʻi ʻia, ua kākau ʻia ka nui o kahi vector ma ka italics, no ka laʻana , F.
2. Hoʻohui ʻana o nā vectors—nā ʻano hana kiʻi
Aia kekahi mau ala e hoʻohui ai i nā vectors ma ke ʻano kiʻi, me ke ʻano Tail-to-tip a me ke ʻano parallelogram.
a. Ke ʻano hana hoʻohui vectors mai ka huelo a i ka piko
Ua ʻike ʻia nā Vectors A a me B. Kūlike ʻo Vector A = 3 cm me ke axis-x (ma ka hikina). Hoʻokumu ʻo Vector B = 2 cm i kahi kihi o 30 o i ke axis-x (ma ka hikina ʻākau). E hoʻohui iā A a me B ma ke ʻano kiʻi me ka hoʻohana ʻana i ke ʻano Tail-to-tip. a) R = A + B b) R = A – B
Ua ana ʻia ka nui o ka vector hopena (R) me ka hoʻohana ʻana i kahi lula. Ua ana ʻia ke kuhikuhi o ka vector hopena me ka hoʻohana ʻana i kahi protractor.
Nā vectors i ʻike ʻia ʻo A, B, a me C. Kūlike ka Vector A = 3 cm me ke axis-x (ma ka hikina). Hoʻokumu ka Vector B = 2 cm i kahi kihi o 30 o i ke axis-x (ma ka hikina ʻākau). Hoʻokumu ka Vector C = 1 cm i kahi kihi o 60 o i ke axis-x (ma ka hikina ʻākau). E hoʻohui iā A, B, a me C ma ke ʻano kiʻi me ka hoʻohana ʻana i ke ʻano Tail-to-tip.
a) R = A + B + C
b) R = A – B – C
Ua ana ʻia ke kuhikuhi o ka vector hopena (R) me ka hoʻohana ʻana i kahi lula. Ua ana ʻia ke kuhikuhi o ka vector hopena me ka hoʻohana ʻana i kahi protractor.
b. Ke ʻano hana paralelogram
Nā vectors i ʻike ʻia ʻo A, B, a me C. Kūlike ka Vector A = 3 cm me ke axis-x (ma ka hikina). Hoʻokumu ka Vector B = 2 cm i kahi kihi o 30 o i ke axis-x (ma ka hikina ʻākau). Hoʻokumu ka Vector C = 1 cm i kahi kihi o 60 o i ke axis-x (ma ka hikina ʻākau). E hoʻohui iā A, B, a me C ma ke ʻano kiʻi me ka hoʻohana ʻana i kahi parallelogram.
a) R = A + B
b) R = A – B
c) R = A + B + C
d) R = A – B – C
Ua ana ʻia ke kuhikuhi o ka vector hopena (R) me ka hoʻohana ʻana i kahi lula. Ua ana ʻia ke kuhikuhi o ka vector hopena me ka hoʻohana ʻana i kahi protractor.
3. Hoʻohui ʻia o nā vectors - aloiloi hana
ʻO ka hoʻoholo ʻana i ka nui a me ke kuhikuhi o ka vector hopena me ke ʻano kiʻi kekahi ala. ʻO ka pololei o nā hopena e pili ana i kou pololei a me ka pololei i ke kaha kiʻi ʻana a me ka heluhelu ʻana i ka unahi. ʻOi aku ka pololei o ka loaʻa ʻana o ka nui a me ke kuhikuhi o ka vector hopena ma o nā helu makemakika.
a. Ke helu nei i ka huina o 2 mau vectors me ka hoʻohana ʻana i ke kānāwai cosine
ʻO ke ʻano hana no ka hoʻoholo ʻana i ka nui o ka vector hopena:
ʻO ke ʻano hana no ka hoʻoholo ʻana i ke kuhikuhi o ka vector hopena:
C = vector hopena
A = vector 1
B = vector 2
cos ∠(A, B) = kihi i hoʻokumu ʻia e nā vectors A a me B
Laʻana pilikia 1:
Hoʻokumu ʻo F 1 = 2 N i kahi kihi o 30 e pili ana i ke axis-x, F 2 = 3 Hoʻokumu ʻo N i kahi kihi o 60 o e pili ana i ke axis-x, θ = 30 o.
Laʻana pilikia 2:
Kūlike ʻo F 1 = 2 N me ke axis-x, hana ʻo F 2 = 3 N i kahi kihi o 90 o a puni ke axis-x, θ = 90 o.
b. Hoʻohui i nā vectors e nā ʻāpana
E nānā hou i kahi vector F e hana ana i kahi kihi e pili ana i ka axis-x, e like me ka mea i hōʻike ʻia ma ke kiʻi ma lalo. ʻO F x a me F y nā vectors ʻāpana o ka vector F.
Hoʻoholo ʻia ka nui o ka vector ʻāpana me ka hoʻohana ʻana i ke ʻano:
F x = F cos θ
F y = F sin θ
E nānā hou i nā vectors ʻelua F1 a me F2 ka mea e hana ana i kahi kihi e pili ana i ka axis-x, e like me ka mea i hōʻike ʻia ma ke kiʻi ma lalo nei. F1x a me F1y he mau ʻāpana o ka vector F1, no laila F2x a me F2y he mau ʻāpana o ka vector F2.
Hoʻoholo ʻia ka vector ʻāpana me ka hoʻohana ʻana i ke ʻano:
F1x =F1 cos θ
F 1y = F 1 sin θ
F 2x = F 2 cos θ
F 1y = F 1 sin θ
Hoʻohui i nā vectors ʻāpana:
Fx = F1x + F2x
Fy = F 1y + F 2y
Hoʻoholo ʻia ka vector hopena me ka hoʻohana ʻana i ke ʻano:
Hoʻoholo ʻia ke kuhikuhi o ka vector hopena me ka hoʻohana ʻana i ke ʻano:
Laʻana pilikia 1:
E hoʻoholo i nā ʻāpana o ka vector F nona ka nui he 20 N a hana i kahi kihi o 30 o a puni ka axis-x.
F x = (20 N)(cos 30 o ) = 17 N
F y = (20 N)(hala 30 o ) = 10 N
Laʻana pilikia 2:
Hoʻokumu ʻo F 1 = 20 N i kahi kihi o 30 o e pili ana i ke axis-x a hoʻokumu ʻo F 2 = 15 N i kahi kihi o 180o e pili ana i ke axis-x. E hoʻoholo i ka nui a me ke kuhikuhi o ka vector hopena.
F 1x = (20 N)(cos 30 o ) = 17 N
F 2x = F 1 = – 15 N
F 1y = (20 N)(hala 30 o ) = 10 N
F 2y = 0
Hoʻohui i nā vectors ʻāpana:
F x = F 1x – F 2x = 17 N – 15 N = 2 N
Fy = 10 N
ʻO ka vector hopena:
F 2 = F x 2 + F y 2 = 2 2 + 10 2 = 4 + 100 = 104
F = 10.2N
Ke kuhikuhi o ka vector hopena:
θ = tan −15 = 78.7 o e pili ana i ke axis-x
Laʻana pilikia 3:
F1, F2, a me F3 he 20 N, 30 N a me 40 N. F1 hana i kahi kihi o 60o e pili ana i ke axis-x, F2 hana i kahi kihi o 150o e pili ana i ke axis-x, a me F3 hana i kahi kihi o 315o e pili ana i ke axis-x. E hoʻoholo i ka nui a me ke kuhikuhi o ka vector hopena.
F 1x = (20 N)(cos 60 o ) = 10 N
F 1y = (20 N)(hala 60 o ) = 17 N
F 2x = (30 N)(cos 30 o ) = -26 N
F 2y = (30 N)(hala 30 o ) = 15 N
F 3x = (40 N)(cos 45 o ) = 28 N
F 3y = (40 N)(hala 45 o ) = -28 N
Ke hoʻohui nei i ka vector ʻāpana:
ʻO F x = F 1x – F 2x + F 3x = 10 N – 26 N + 28 N = 12 N
F y = F 1y + F 2y – F 3y = 17 N + 15 N – 28 N = 4 N
Vector hopena:
F 2 = F x 2 + F y 2 = 12 2 + 4 2 = 144 + 16 = 160
F = 13N
Ke kuhikuhi o ka vector hopena:
θ = tan −1 0.3 = 17 o e pili ana i ke axis-x
