Chitsanzo cha Funso Lokambirana za Kuwonjezera Ma Vector Awiri Pogwiritsa Ntchito Njira ya Parallelogram
Kuwonjezera maveketa ndi lingaliro lofunika kwambiri mu fizikisi ndi masamu, lomwe nthawi zambiri limagwiritsidwa ntchito pofotokoza zochitika zachilengedwe ndi mavuto a tsiku ndi tsiku. Pali njira zingapo zowonjezera maveketa awiri, imodzi mwa izo ndi njira ya parallelogram. Njirayi sikuti ndi yongomveka bwino komanso imapereka chithunzi champhamvu cha momwe maveketa awiri amagwirizanirana kuti apange veketa yotsatira. M'nkhaniyi, tiwona zitsanzo zingapo za kuwonjezera maveketa pogwiritsa ntchito njira ya parallelogram, pamodzi ndi mayankho awo.
Kodi Vector ndi chiyani?
Tisanalowe mu zitsanzo za mavuto, tiyenera kumvetsetsa tanthauzo loyambira la vekitala. Vekitala ndi kuchuluka komwe kuli ndi kukula (kutalika) ndi komwe kumayang'aniridwa. Zitsanzo zakale za vekitala zimaphatikizapo liwiro, kuthamanga, mphamvu, ndi kusamuka. Vekitala ikhoza kuimiridwa ngati zigawo zake (i, j, k) mu ma coordinates a Cartesian kapena ngati kutalika kwake ndi komwe kumayang'aniridwa (ngodya).
Njira ya Paralelogram
Njira ya parallelogram ndi njira imodzi yowonjezera mavekitala awiri. Mu njira iyi, timayimira mavekitala awiri ngati mbali ziwiri za parallelogram. Vekitala yotsatira ndi diagonal ya parallelogram kuyambira poyambira mavekitala awiri. Mwa masamu, ngati tili ndi mavekitala awiri \(\vec{A}\) ndi \(\vec{B}\), zotsatira zake ndi \( \vec{R} = \vec{A} + \vec{B} \).
Njira yogwiritsira ntchito njira ya parallelogram ndi iyi:
1. Jambulani vekitala \(\vec{A}\) kuchokera poyambira.
2. Kuchokera kumapeto kwa vekitala \(\vec{A}\), jambulani vekitala \(\vec{B}\).
3. Jambulani mzere wofanana ndi vekitala \(\vec{B}\) kuchokera poyambira \(\vec{A}\).
4. Jambulani mzere wofanana ndi vekitala \(\vec{A}\) kuchokera kumapeto kwa vekitala \(\vec{B}\).
5. Jambulani chopingasa kuchokera poyambira kupita ku ngodya ina kuti mupeze vekitala yotsatira \(\vec{R}\).
Mafunso ndi Kukambirana Zitsanzo
Funso 1
Tiyerekeze kuti tili ndi mavekitala awiri \(\vec{A}\) ndi \(\vec{B}\):
– \(\vec{A}\) ili ndi kutalika (kukula) kwa mayunitsi 5 ndi njira ya 0° (kapena motsatira x-axis yabwino),
– \(\vec{B}\) ili ndi utali wa mayunitsi atatu ndi njira ya 90° (kapena motsatira mzere wa y wabwino).
Kodi phindu lowonjezera ma vector awiriwa pogwiritsa ntchito njira ya parallelogram ndi lotani?
Kukambirana:
1. Jambulani vekitala \(\vec{A}\) motsatira mzere wa x wabwino wokhala ndi utali wa mayunitsi 5.
2. Kuchokera kumapeto kwa vekitala \(\vec{A}\), jambulani vekitala \(\vec{B}\) motsatira mzere wa y wabwino wokhala ndi utali wa mayunitsi atatu.
3. Kuchokera poyambira \(\vec{A}\), jambulani mzere wofanana ndi \(\vec{B}\).
4. Kuchokera kumapeto kwa \(\vec{B}\), jambulani mzere wofanana ndi \(\vec{A}\).
5. Zotsatira zake ndi parallelogram yokhala ndi diagonal yomwe ndi vekitala yotsatira \(\vec{R}\).
Popeza \(\vec{A}\) ndi \(\vec{B}\) ndi olunjika kwa wina ndi mnzake, tingagwiritse ntchito chiphunzitso cha Pythagorean kuti tiwerengere kutalika kwa vekitala yotsatira:
\[ R = \sqrt{A^2 + B^2} = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34} \pafupifupi 5.83 \]
Kulunjika kwa vekitala yotuluka kumatha kuwerengedwa pogwiritsa ntchito trigonometry. Ngati \(\theta\) ndi ngodya pakati pa zotsatira ndi \(\vec{A}\):
\[ \tan(\theta) = \frac{B}{A} = \frac{3}{5} \]
kotero:
\[ \theta = \tan^{-1}\left(\frac{3}{5}\right) \approx 30.96^\circ \]
Motero, vekitala yotsatira \(\vec{R}\) ili ndi kukula kwa pafupifupi mayunitsi 5.83 ndi njira ya pafupifupi 30.96° kuchokera ku \(\vec{A}\).
Funso 2
Ma vekitala awiri \(\vec{C}\) ndi \(\vec{D}\) aperekedwa motere:
– \(\vec{C}\) yokhala ndi utali wa mayunitsi 4 ndi njira ya 45°.
– \(\vec{D}\) yokhala ndi utali wa mayunitsi 6 ndi njira ya 120°.
Dziwani vekitala yotsatira \(\vec{R}\) kuchokera pakuwonjezera mavekitala awiriwo.
Kukambirana:
Kuti muwonjezere ma vector awiri omwe sali olunjika kwa wina ndi mnzake kapena okhala ndi mawonekedwe osiyanasiyana, mutha kugwiritsa ntchito zigawo za Cartesian.
1. Gawani \(\vec{C}\) ndi \(\vec{D}\) m'zigawo za x ndi y.
Kwa \(\vec{C}\):
\[ C_x = C \cos(45^\circ) = 4 \cos(45^\circ) = 4 \cdot \frac{\sqrt{2}}{2} = 2\sqrt{2} \pafupifupi 2.83 \]
\[ C_y = C \sin(45^\circ) = 4 \sin(45^\circ) = 4 \cdot \frac{\sqrt{2}}{2} = 2\sqrt{2} \pafupifupi 2.83 \]
Kwa \(\vec{D}\):
\[ D_x = D \cos(120^\circ) = 6 \cos(120^\circ) = 6 \cdot (-\frac{1}{2}) = -3 \]
\[ D_y = D \sin(120^\circ) = 6 \sin(120^\circ) = 6 \cdot \frac{\sqrt{3}}{2} = 3\sqrt{3} \pafupifupi 5.20 \]
2. Onjezani zigawo za x ndi y za ma vekitala onse awiri:
\[ R_x = C_x + D_x = 2.83 + (-3) = -0.17 \]
\[ R_y = C_y + D_y = 2.83 + 5.20 = 8.03 \]
3. Werengani kukula ndi njira ya vekitala yotuluka \(\vec{R}\):
\[ R = \sqrt{R_x^2 + R_y^2} = \sqrt{(-0.17)^2 + 8.03^2} = \sqrt{0.03 + 64.48} = \sqrt{64.51} \pafupifupi 8.03 \]
\[ \theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) = \tan^{-1}\left(\frac{8.03}{-0.17}\right) \approx \tan^{-1}(-47.24) \]
Popeza zotsatira zake ndi zoipa, timawonjezera 180° kuti tipeze ngodya mu dongosolo lolondola la quadrant:
\[ \theta \approx \tan^{-1}(47.24) + 180^\circ \approx 271.93^\circ \]
Kotero, vekitala yotsatila \(\vec{R}\) ili ndi kukula kwa pafupifupi mayunitsi 8.03 ndi njira ya pafupifupi 271.93°, kapena tinganene pafupifupi 91.93° kuchokera ku x-axis yoyipa mu quadrant yachinayi.
Kutseka
Njira ya parallelogram ndi njira yothandiza komanso yowoneka bwino yowonjezerera mavekitala awiri. Ngakhale njira iyi ingawoneke yosavuta kwa mavekitala osavuta, ndikofunikira kumvetsetsa kuti kwa mavekitala ovuta kwambiri, nthawi zambiri timafunika kugwiritsa ntchito zigawo za Cartesian ndi njira zapamwamba kwambiri za algebraic kuti tipeze zotsatira zolondola. Tikukhulupirira kuti zitsanzo zomwe zili pamwambapa zikupereka chithunzi chomveka bwino cha momwe njira iyi ingagwiritsidwire ntchito pazochitika zosiyanasiyana.
