1. Masalani bayon qilamiz: Muntazam 6 burchakli ko'pburchakning har bir uchiga qarab yo'nalgan kuchlar berilgan: 1, 3, 5, 7, 9, 13 N. Biz teng ta'sir etuvchi va muvozanatlovchi kuchning miqdori va yo'nalishini topamiz. 2. Muntazam 6 burchakda har bir burchak orasidagi burchak $60^\circ$ ga teng. 3. Har bir kuchni vektor sifatida ifodalaymiz. Kuchlarning yo'nalishi burchak markazidan ko'pburchakning uchlariga qarab bo'lgani uchun, kuch $i$-chi kuchning yo'nalishi $\theta_i = 60^\circ \times (i-1)$, $i=1,2,...,6$. 4. Har bir kuchning $x$ va $y$ komponentlarini hisoblaymiz: $$F_{x_i} = F_i \cos(\theta_i), \quad F_{y_i} = F_i \sin(\theta_i)$$ 5. Kuchlar: $F_1=1$, $F_2=3$, $F_3=5$, $F_4=7$, $F_5=9$, $F_6=13$ 6. Burchaklar (gradusda): $\theta_1=0^\circ$, $\theta_2=60^\circ$, $\theta_3=120^\circ$, $\theta_4=180^\circ$, $\theta_5=240^\circ$, $\theta_6=300^\circ$ 7. Hisoblaymiz: $$F_x = \sum_{i=1}^6 F_i \cos(\theta_i) = 1\cos0^\circ + 3\cos60^\circ + 5\cos120^\circ + 7\cos180^\circ + 9\cos240^\circ + 13\cos300^\circ$$ $$= 1 + 3 \times 0.5 + 5 \times (-0.5) + 7 \times (-1) + 9 \times (-0.5) + 13 \times 0.5$$ $$= 1 + 1.5 - 2.5 - 7 - 4.5 + 6.5 = (1 + 1.5 + 6.5) - (2.5 + 7 + 4.5) = 9 - 14 = -5$$ 8. $y$ komponenti: $$F_y = \sum_{i=1}^6 F_i \sin(\theta_i) = 1\sin0^\circ + 3\sin60^\circ + 5\sin120^\circ + 7\sin180^\circ + 9\sin240^\circ + 13\sin300^\circ$$ $$= 0 + 3 \times \frac{\sqrt{3}}{2} + 5 \times \frac{\sqrt{3}}{2} + 0 + 9 \times (-\frac{\sqrt{3}}{2}) + 13 \times (-\frac{\sqrt{3}}{2})$$ $$= \frac{\sqrt{3}}{2} (3 + 5 - 9 - 13) = \frac{\sqrt{3}}{2} (-14) = -7\sqrt{3}$$ 9. Teng ta'sir etuvchi kuchning miqdori: $$F = \sqrt{F_x^2 + F_y^2} = \sqrt{(-5)^2 + (-7\sqrt{3})^2} = \sqrt{25 + 49 \times 3} = \sqrt{25 + 147} = \sqrt{172} \approx 13.114$$ 10. Yo'nalishi (burchak $\alpha$) $x$ o'qiga nisbatan: $$\alpha = \arctan\left(\frac{F_y}{F_x}\right) = \arctan\left(\frac{-7\sqrt{3}}{-5}\right) = \arctan\left(\frac{7\sqrt{3}}{5}\right)$$ Ikkala $F_x$ va $F_y$ manfiy bo'lgani uchun, burchak uchinchi chorakda: $$\alpha = 180^\circ + \arctan\left(\frac{7\sqrt{3}}{5}\right) \approx 180^\circ + 69.295^\circ = 249.295^\circ$$ 11. Muvozanatlovchi kuch teng ta'sir etuvchi kuchga qarama-qarshi yo'nalishda bo'ladi, shuning uchun uning miqdori $13.114$ N va yo'nalishi $249.295^\circ - 180^\circ = 69.295^\circ$. Javob: Teng ta'sir etuvchi kuch $\approx 13.114$ N, yo'nalishi $249.3^\circ$; muvozanatlovchi kuch $\approx 13.114$ N, yo'nalishi $69.3^\circ$.