Ukwengezwa kwamaVector

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 ukufuduka, ukusheshisa, i-impulse, umfutho, njll. Ungakuqonda kahle uma ufunda izihloko ezihlobene nalezo zibalo.

b. Dweba i-vektha

Ivektha ikhonjiswa ngomcibisholo. Umcibisholo uhlala udonswa ukuze ukhombe ohlangothini lwevektha. Ubude bomcibisholo buchazwa njengobukhulu bevektha. Isibonelo, esithombeni sevektha yamandla (F) enobukhulu obungu-2 N obheke ngaso enyakatho-mpumalanga noma ku-45^o mayelana ne-x-axis.

c. Rizinduna okwe winani lokurekhoda le-vector

Ekubhaleni i-vector, uma usebenzisa umbhalo wesandla, uphawu lwe-vector luvame ukubhalwa ngamagama aqoshiwe kusetshenziswa izinhlamvu ezinkulu, futhi ngaphezulu kwalo kudinga ukungezwa ngomcibisholo. Ezincwadini eziphrintiwe, izimpawu ze-vector zibhalwa ngezinhlamvu ezinkulu ngokugqamile, isibonelo, F. Ngobukhulu bevektha, uma sisebenzisa umbhalo wesandla khona-ke ubukhulu bevektha buyabhalwa, isibonelo, |F|. Ezincwadini ezinyathelisiwe, ubukhulu bevektha bubhalwa ngamagama aqoshiwe, 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

Amavekhtha A no-B ayaziwa. Ivekhtha A = 3 cm ihambisana ne-x-axis (ngasempumalanga). Ivekhtha B = 2 cm yakha i-engeli engu-30^o 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 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). Ivektha C = 1 cm yakha i-engeli engu-60^o 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). Ivektha C = 1 cm yakha i-engeli engu-60^o 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:

F₁ = 2 U-N wakha i-engeli engu-30 mayelana ne-x-axis, u-F₂ = 3 N yakha i-engeli engu-60^o mayelana ne-x-axis, θ = 30^o.

Inkinga yesampula 2:

F₁ = 2 N ihambisana ne-x-axis, F₂ = 3 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._x noF_y ziyizingxenye ze-vector F.

Ubukhulu bevektha yengxenye bunqunywa kusetshenziswa ifomula:

F_x = F cos θ

F_y = F isono θ

Buyekeza amavektha amabili u-F₁ noF₂ okwakha i-engeli ethile mayelana ne-x-axis, njengoba kuboniswe esithombeni esingezansi. F_1x noF_1y yizingxenye ze-vector F₁, ngakho-ke F_2x noF_2y yizingxenye ze-vector F₂.

Ivektha yengxenye inqunywa kusetshenziswa ifomula:

F_1x =F₁ cosθ

F_1y =F₁ isono θ

F_2x =F₂ cosθ

F_1y =F₁ isono 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:

Nquma izingxenye zevektha F ngobukhulu bayo 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:

F₁ = 20 N yakha i-engeli engu-30^o mayelana ne-x-axis kanye ne-F₂ = 15 N yakha i-engeli engu-180o mayelana ne-x-axis. Nquma ubukhulu kanye nesiqondiso sevektha ephumelayo.

F_1x = (20 N)(cos 30^o) = 17 N

F_2x =F₁ = – 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 uN

Ivektha ephumela:

F₂ =F_x² +F_y² = 2² 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, F_2, noF₃ zingama-N angu-20, ama-N angu-30 kanye nama-N angu-40₁ yakha i-engeli engu-60^o mayelana ne-x-axis, F₂ yakha i-engeli engu-150^o mayelana ne-x-axis, kanye ne-F₃ yakha i-engeli engu-315^o 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:

F² =F_x² +F_y² = 12² 4 +² = 144 + 16 = 160

F = 13N

Isiqondiso sevektha ephumela:

θ = tan^-1 = 0.3 17^o mayelana ne-x-axis