Mafunso Okhudza Kuonjezera Vekitala ndi Gawo
Kuwonjezera maveketa ndi njira yofunikira kwambiri mu fizikisi ndi masamu yomwe imagwiritsidwa ntchito kupeza zotsatira za maveketa awiri kapena kuposerapo. Njira yothetsera kuwonjezera maveketa pogwiritsa ntchito zigawo ndi njira yothandiza kwambiri, makamaka pochita ndi maveketa m'magawo awiri kapena atatu. Nkhaniyi ifotokoza lingaliro la kuwonjezera maveketa pogwiritsa ntchito zigawo ndikupereka zitsanzo zingapo za mavuto ndi mayankho.
Lingaliro la Kuwonjezera Vector Yophatikizana
Vekitala iliyonse yomwe ili mu malo amitundu iwiri (2D) ikhoza kugawidwa m'magawo awiri: gawo la x (lopingasa) ndi gawo la y (loyima). Mu magawo atatu (3D), mavekitala ali ndi gawo lina, gawo la z (kuya).
Tiyerekeze kuti tili ndi mavekitala awiri A ndi B. Zigawo za mavekitala awa zitha kufotokozedwa motere:
– Vekitala A ili ndi zigawo \(A_x\) ndi \(A_y\) mu 2D (kapenanso \(A_z\) mu 3D).
– Vekitala B ili ndi zigawo \(B_x\) ndi \(B_y\) mu 2D (kapenanso \(B_z\) mu 3D).
Kuwonjezeredwa kwa ma vector awiriwa kudzapanga vector R yomwe ili ndi zigawo zotsatirazi:
\[ R_x = A_x + B_x \]
\[ R_y = A_y + B_y \]
Kwa ma vector mu 3D, gawo la z ndi ilinso motere:
\[ R_z = A_z + B_z \]
Pambuyo powerengera gawo lililonse la vekitala yotuluka, titha kupeza modulus (kukula) ndi komwe vekitala yotuluka ikupita pogwiritsa ntchito fomula iyi:
\[ |R| = \sqrt{R_x^2 + R_y^2} \] (ya 2D)
Kapena za 3D:
\[ |R| = \sqrt{R_x^2 + R_y^2 + R_z^2} \]
Ndipo komwe vekitala yotsatira ikupita kumatha kudziwika ndi ngodya yopita ku ma axes ogwirizana.
Mafunso ndi Kukambirana Zitsanzo
Funso 1
Popeza ma vector awiri mu ndege yamitundu iwiri:
– A ndi \(5 \, \text{unit}\) kum'mawa.
– B ndi \(3 \, \text{unit}\) kumpoto.
Dziwani vekitala yotsatira R.
Zokambirana
Choyamba, timasintha vekitala kukhala zigawo zake.
– Vekitala A: \(A = (5, 0)\) chifukwa ili ndi gawo la x lokha.
– Vekitala B: \(B = (0, 3)\) chifukwa ili ndi gawo la y lokha.
Nayi chiwerengero cha zigawo:
\[ R_x = A_x + B_x = 5 + 0 = 5 \]
\[ R_y = A_y + B_y = 0 + 3 = 3 \]
Kenako vekitala yotsatira R ndi:
\[ R = (5, 3) \]
Kuti muwerenge kutalika (modulus) kwa vekitala R:
\[ |R| = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34} \pafupifupi 5.83 \]
Kuwongolera kwa vekitala R kumatha kuwerengedwa pogwiritsa ntchito ngodya ya θ kupita ku x-axis:
\[ \tan(\theta) = \frac{R_y}{R_x} = \frac{3}{5} \]
\[ \theta = \arctan\left(\frac{3}{5}\right) \approx 30.96^\circ \]
Motero, vekitala yotsatira R ili ndi kutalika kwa mayunitsi pafupifupi 5.83 ndipo imapanga ngodya ya 30.96° ndi x-axis.
Funso 2
Popeza ma vector awiri m'magawo atatu:
– A ndi \(3\hat{i} + 2\hat{j} + 1\hat{k}\)
– B ndi \(1\hat{i} + 4\hat{j} + 2\hat{k}\)
Dziwani vekitala yotsatira R.
Zokambirana
Choyamba, timazindikira zigawo za vekitala iliyonse:
– Vekitala A: \(A_x = 3\), \(A_y = 2\), \(A_z = 1\).
– Vekitala B : \(B_x = 1\), \(B_y = 4\), \(B_z = 2\).
Nayi chiwerengero cha zigawo:
\[ R_x = A_x + B_x = 3 + 1 = 4 \]
\[ R_y = A_y + B_y = 2 + 4 = 6 \]
\[ R_z = A_z + B_z = 1 + 2 = 3 \]
Kenako vekitala yotsatira R ndi:
\[ R = (4, 6, 3) \]
Kuti muwerenge kutalika (modulus) kwa vekitala R:
\[ |R| = \sqrt{4^2 + 6^2 + 3^2} = \sqrt{16 + 36 + 9} = \sqrt{61} \pafupifupi 7.81 \]
Kuwongolera kwa vekitala R poyerekeza ndi ma axes a x, y, ndi z kumatha kuwerengedwa pogwiritsa ntchito cosine ya director:
\[ \cos(\alpha) = \frac{R_x}{|R|} = \frac{4}{7.81} \pafupifupi 0.512 \]
\[ \alpha = \arccos(0.512) \pafupifupi 59.50^\circ \]
\[ \cos(\beta) = \frac{R_y}{|R|} = \frac{6}{7.81} \pafupifupi 0.768 \]
\[ \beta = \arccos(0.768) \pafupifupi 39.50^\circ \]
\[ \cos(\gamma) = \frac{R_z}{|R|} = \frac{3}{7.81} \pafupifupi 0.384 \]
\[ \gamma = \arccos(0.384) \pafupifupi 67.64^\circ \]
Motero, vekitala yotsatira R ili ndi kutalika kwa mayunitsi pafupifupi 7.81 ndipo malangizo ake poyerekeza ndi ma axes a x, y, ndi z ndi 59.50°, 39.50°, ndi 67.64°.
Funso 3
Popeza ma vector awiri:
– P ili ndi kukula kwa mayunitsi 4 ndipo imapanga ngodya ya 45° ku x-axis yabwino.
– Q ili ndi magnitude a mayunitsi 6 ndipo imapanga ngodya ya 120° ku x-axis yabwino.
Dziwani vekitala yotsatira R.
Zokambirana
Choyamba, timagawa vekitala m'zigawo zake za x ndi y:
– Vekitala P : \(P_x = 4\cos(45^\circ) = 4 \cdot \frac{\sqrt{2}}{2} \approx 2.83\), \(P_y = 4\sin(45^\circ) = 4 \cdot \frac{\sqrt{2}}{2} \approx 2.83\).
– Vekitala Q : \(Q_x = 6\cos(120^\circ) = 6 \cdot \left(-\frac{1}{2}\right) = -3\), \(Q_y = 6\sin(120^\circ) = 6 \cdot \frac{\sqrt{3}}{2} \approx 5.2\).
Nayi chiwerengero cha zigawo:
\[ R_x = P_x + Q_x = 2.83 – 3 = -0.17 \]
\[ R_y = P_y + Q_y = 2.83 + 5.2 = 8.03 \]
Kenako, vekitala yotsatira R ndi:
\[ R = (-0.17, 8.03) \]
Kuti muwerenge kutalika (modulus) kwa vekitala R:
\[ |R| = \sqrt{(-0.17)^2 + 8.03^2} = \sqrt{0.0289 + 64.48} = \sqrt{64.509} \pafupifupi 8.03 \]
Malangizo a vekitala R:
\[ \tan(\theta) = \frac{R_y}{R_x} = \frac{8.03}{-0.17} = -47.24 \]
\[ \theta = \arctan(-47.24) \pafupifupi -88.99^\circ \]
Komabe, ngodya iyi imayesedwa pafupi ndi x-axis yoyipa, kotero ngodya yeniyeni munkhaniyi ndi iyi:
\[ 180^\circ – 88.99^\circ \pafupifupi 91.01^\circ \]
Motero, vekitala yotsatira R ili ndi kutalika kwa mayunitsi pafupifupi 8.03 ndipo imapanga ngodya ya 91.01° ndi x-axis yabwino.
Nkhaniyi yafotokoza za kuwonjezera mavekitala pogwiritsa ntchito zigawo, popereka zitsanzo zingapo za mavuto ndi mayankho. Njira yogwiritsira ntchito zigawo ndi yothandiza kwambiri popangitsa kuti kuwerengera kukhale kosavuta komanso kupereka njira yokhazikika yothetsera mavuto a mavekitala pogwiritsa ntchito masamu a malo.
