Isihloko mayelana nokwengezwa kwama-Vector
1. Ubuningi be-vector kanye ne-scalar
Ngaphezu kobuningi obuyisisekelo nobuvelayo, ubuningi obuphathekayo busengahlukaniswa ngezinhlobo ezimbili, okungukuthi ubuningi be-scalar kanye nobuningi be-vector. Ubuningi obufana nobunzima, ibanga, isikhathi kanye nomthamo, buyinani le-scalar, ubuningi obunobukhulu kodwa obungenalo isiqondiso. Ngenkathi ubukhulu obufana nokufuduka, ijubane, ukusheshisa, kanye namandla kuyinani le-vector, ubuningi obunobukhulu futhi obunesiqondiso.
a. Umehluko phakathi kwe-s calar kanye nenani le-vector
Uma uthi isisindo sebhola singamagremu angu-400, lesi sitatimende sanele ukuthi wazi isisindo sebhola. Awudingi isiqondiso ukuze uthole isisindo sebhola. Ngokufanayo nesikhathi, izinga lokushisa, ivolumu, ubuningi, njll. Kunezinhlobo eziningana zomzimba ezingeke zichazwe ngobukhulu kuphela. Uma uthi ingane inyakaza ize ifike kumamitha ayi-100, khona-ke lesi sitatimende asanele. Ungase ubuze, inyakaze kuphi? Ingabe isenyakatho, eningizimu, empumalanga, noma entshonalanga? Ngokufanayo, uma uthi ucindezela itafula ngamandla angu-200 N.
Ushayela kuphi? Nokho, amanani anjalo abizwa ngokuthi amanani e-vector, adinga incazelo yobukhulu kanye nesiqondiso. Izibonelo zamanani e-vector yi-displacement, i-acceleration, i-impulse, i-momentum , njll. Ungakuqonda ngokucacile lapho ufunda izihloko ezihlobene nalawo manani.
b. Dweba i-vektha
Ivektha ikhonjiswa ngomcibisholo. Umcibisholo uhlala udwetshwa ukuze ukhombe ohlangothini lwevektha. Ubude bomcibisholo buchazwa njengobukhulu bevektha. Isibonelo, esithombeni sevektha yamandla (F) enobukhulu obungu-2 N okuqondisa kwayo kubheke enyakatho-mpumalanga noma ku-45 o mayelana ne-x-axis.
c. Imithetho yokubhala inani le- vector
Ekubhaleni i-vector, uma usebenzisa umbhalo wesandla, uphawu lwe-vector luvame ukubhalwa ngama-italics kusetshenziswa izinhlamvu ezinkulu, futhi ngaphezulu kwalo kudinga ukungezwa ngomcibisholo. Ezincwadini eziphrintiwe, izimpawu ze-vector zibhalwa ngezinhlamvu ezinkulu ngokugqamile, isibonelo, F. Ngobukhulu be-vector, uma sisebenzisa umbhalo wesandla khona-ke ubukhulu be-vector buyabhalwa, isibonelo, |F|. Ezincwadini eziphrintiwe, ubukhulu be-vector bubhalwa ngama-italics, isibonelo , F.
2. Ukwengezwa kwamavektha—izindlela zokudweba
Kunezindlela eziningana zokwengeza ama-vector ngemidwebo, okuhlanganisa indlela ye-Tail-to-tip kanye nendlela ye-parallelogram.
a. Indlela yokwengeza ama-vector ukusuka komsila kuya komucu
Amavektha A no-B ayaziwa. Ivektha A = 3 cm ihambisana ne-x-axis (ngasempumalanga). Ivektha B = 2 cm yakha i-engeli engu-30 o kuya ku-x-axis (ngasenyakatho-mpumalanga). Engeza u-A no-B ngemidwebo usebenzisa indlela ye-Tail-to-tip. a) R = A + B b) R = A – B
Ubukhulu bevektha ephumayo (R) kulinganiswa kusetshenziswa irula. Isiqondiso sevektha ephumayo kulinganiswa kusetshenziswa i-protractor.
Amavekhtha aziwayo u-A, u-B, no-C. Ivekhtha A = 3 cm ihambisana ne-x-axis (ngasempumalanga). Ivekhtha B = 2 cm yakha i-engeli engu-30 o kuya ku-x-axis (ngasenyakatho-mpumalanga). Ivekhtha C = 1 cm yakha i-engeli engu-60 o kuya ku-x-axis (ngasenyakatho-mpumalanga). Engeza u-A, u-B, no-C ngemidwebo usebenzisa indlela ye-Tail-to-tip.
a) R = A + B + C
b) R = A – B – C
Ivektha ephumayo (R) ilinganiswa kusetshenziswa irula. Isiqondiso sevektha ephumayo silinganiswa kusetshenziswa i-protractor.
b. Indlela ye-Parallelogram
Amavekhtha aziwayo u-A, u-B, no-C. Ivekhtha u-A = 3 cm ihambisana ne-x-axis (ngasempumalanga). Ivekhtha u-B = 2 cm yakha i-engeli engu-30 o kuya ku-x-axis (ngasenyakatho-mpumalanga). Ivekhtha u-C = 1 cm yakha i-engeli engu-60 o kuya ku-x-axis (ngasenyakatho-mpumalanga). Engeza u-A, u-B, no-C ngemidwebo usebenzisa i-parallelogram.
a) R = A + B
b) R = A – B
c) R = A + B + C
d) R = A – B – C
Ivektha ephumayo (R) ilinganiswa kusetshenziswa irula. Isiqondiso sevektha ephumayo silinganiswa kusetshenziswa i-protractor.
3. Ukwengezwa kwamavektha – ai-nalytical indlela
Ukunquma ubukhulu kanye nesiqondiso sevektha ephumayo ngendlela yesithombe kuyindlela eyodwa. Ukunemba kwemiphumela kuncike ekunembeni nasekuqondeni kwakho ekudwebeni nasekufundeni isikali. Ubukhulu kanye nesiqondiso sevektha ephumayo kutholakala ngokunembile ngokubala kwezibalo.
a. Ukubala isamba samavektha amabili kusetshenziswa umthetho we-cosine
Ifomula yokunquma ubukhulu bevektha ephumela:
Ifomula yokunquma isiqondiso sevektha ephumela:
C = i-vector ephumela
A = i-vektha 1
B = i-vektha 2
i-cos ∠(A, B) = i-engeli eyakhiwe yi-vectors A kanye ne-B
Inkinga yesampula 1:
F 1 = 2 U-N wakha i-engeli engu-30 mayelana ne-x-axis, U-F 2 = 3 U-N wakha i-engeli engu-60 o mayelana ne-x-axis, θ = 30 o.
Inkinga yesampula 2:
I-F 1 = 2 I-N ihambisana ne-x-axis, i-F 2 = 3 I-N yakha i-engeli engu-90 o mayelana ne-x-axis, θ = 90 o.
b. Ukwengeza amavektha ngezingxenye
Buyekeza i-vector F eyakha i-engeli ethile mayelana ne-x-axis, njengoba kuboniswe esithombeni esingezansi. I-F x kanye ne-F y ziyi-component vectors ze-vector F.
Ubukhulu bevektha yengxenye bunqunywa kusetshenziswa ifomula:
F x = F cos θ
F y = F sin θ
Buyekeza amavektha amabili u-F1 noF2 okwakha i-engeli ethile mayelana ne-x-axis, njengoba kuboniswe esithombeni esingezansi. F1x noF1y yizingxenye ze-vector F1, ngakho-ke F2x noF2y yizingxenye ze-vector F2.
Ivektha yengxenye inqunywa kusetshenziswa ifomula:
F1x =F1 cosθ
F 1y = F 1 sin θ
F 2x = F 2 cos θ
F 1y = F 1 sin θ
Ukungeza amavekhtha ezingxenye:
F x = F 1x + F 2x
F y = F 1y + F 2y
Ivektha ephumelayo inqunywa kusetshenziswa ifomula:
Isiqondiso sevektha ephumelayo sinqunywa kusetshenziswa ifomula:
Inkinga yesampula 1:
Thola izingxenye zevektha u-F ngobukhulu obungu-20 N bese wakha i-engeli engu-30 o mayelana ne-x-axis.
F x = (20 N)(cos 30 o ) = 17 N
F y = (20 N)(isono 30 o ) = 10 N
Inkinga yesampula 2:
F 1 = 20 N yakha i-engeli engu-30 o mayelana ne-x-axis kanye F 2 = 15 N yakha i-engeli engu-180 o mayelana ne-x-axis. Nquma ubukhulu kanye nesiqondiso sevektha ephumelayo.
F 1x = (20 N)(cos 30 o ) = 17 N
F 2x = F 1 = – 15 N
F 1y = (20 N)(isono 30 o ) = 10 N
F 2y = 0
Ukungeza amavekhtha ezingxenye:
F x = F 1x – F 2x = 17 N – 15 N = 2 N
F y = 10 N
Ivektha ephumela:
F 2 = F x 2 + F y 2 = 2 2 + 10 2 = 4 + 100 = 104
F = 10.2N
Isiqondiso sevektha ephumela:
θ = tan −15 = 78.7 o mayelana ne-x-axis
Inkinga yesampula 3:
F1, F2, noF3 zingama-N angu-20, ama-N angu-30 kanye nama-N angu-401 yakha i-engeli engu-60o mayelana ne-x-axis, F2 yakha i-engeli engu-150o mayelana ne-x-axis, kanye ne-F3 yakha i-engeli engu-315o mayelana ne-x-axis. Nquma ubukhulu kanye nesiqondiso sevektha ephumelayo.
F 1x = (20 N)(cos 60 o ) = 10 N
F 1y = (20 N)(isono 60 o ) = 17 N
F 2x = (30 N)(cos 30 o ) = -26 N
F 2y = (30 N)(isono 30 o ) = 15 N
F 3x = (40 N)(cos 45 o ) = 28 N
F 3y = (40 N)(isono 45 o ) = -28 N
Ukungeza i-component vector:
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
Ivektha ephumela:
F 2 = F x 2 + F y 2 = 12 2 + 4 2 = 144 + 16 = 160
F = 13N
Isiqondiso sevektha ephumela:
θ = tan −1 0.3 = 17 o mayelana ne-x-axis
