Masalalar — Qattiq Jism Mexanikasi

30 ta masala: osondan qiyinga tartiblangan.

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Boshlang'ich (1-10)

Masala 1: Burchak tezlik

G'ildirak 600 RPM (aylanish/minut) bilan aylanmoqda. Burchak tezligini rad/s da toping.

Yechim

$\omega = 600 \times \frac{2\pi}{60} = 600 \times \frac{\pi}{30} = 20\pi \approx 62.8 \text{ rad/s}$

$\boxed{\omega = 20\pi \approx 62.8 \text{ rad/s}}$

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Masala 2: Chiziqli va burchak tezlik

Radiusi 0.3 m bo'lgan g'ildirak 10 rad/s bilan aylanmoqda. Chetidagi nuqtaning chiziqli tezligini toping.

Yechim

$v = r\omega = 0.3 \times 10 = 3 \text{ m/s}$

$\boxed{v = 3 \text{ m/s}}$

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Masala 3: Burchak tezlanish

Motor 0 dan 3000 RPM gacha 5 sekundda tezlashadi. Burchak tezlanishni toping.

Yechim

$\omega_0 = 0$

$\omega = 3000 \times \frac{2\pi}{60} = 100\pi$ rad/s

$\alpha = \frac{\omega - \omega_0}{t} = \frac{100\pi - 0}{5} = 20\pi \approx 62.8 \text{ rad/s}^2$

$\boxed{\alpha = 20\pi \approx 62.8 \text{ rad/s}^2}$

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Masala 4: Inertsiya momenti — disk

Massasi 2 kg, radiusi 0.2 m bo'lgan diskning inertsiya momentini toping.

Yechim

$I = \frac{1}{2}MR^2 = \frac{1}{2} \times 2 \times 0.2^2 = \frac{1}{2} \times 2 \times 0.04 = 0.04 \text{ kg·m}^2$

$\boxed{I = 0.04 \text{ kg·m}^2}$

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Masala 5: Inertsiya momenti — tayoq

Uzunligi 1 m, massasi 3 kg bo'lgan tayoqning markazidan o'tuvchi o'q uchun inertsiya momentini toping.

Yechim

$I = \frac{1}{12}ML^2 = \frac{1}{12} \times 3 \times 1^2 = 0.25 \text{ kg·m}^2$

$\boxed{I = 0.25 \text{ kg·m}^2}$

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Masala 6: Aylantiruvchi moment

Uzunligi 0.5 m bo'lgan kalit bilan 40 N kuch qo'yildi. Aylantiruvchi momentni toping.

Yechim

$\tau = F \times r = 40 \times 0.5 = 20 \text{ N·m}$

$\boxed{\tau = 20 \text{ N·m}}$

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Masala 7: Aylanma harakat tenglamasi

$I = 0.5$ kg·m², $\tau = 10$ N·m. Burchak tezlanishni toping.

Yechim

$\alpha = \frac{\tau}{I} = \frac{10}{0.5} = 20 \text{ rad/s}^2$

$\boxed{\alpha = 20 \text{ rad/s}^2}$

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Masala 8: Aylanma kinetik energiya

Disk $I = 0.1$ kg·m², $\omega = 50$ rad/s. Kinetik energiyasini toping.

Yechim

$K = \frac{1}{2}I\omega^2 = \frac{1}{2} \times 0.1 \times 50^2 = 0.05 \times 2500 = 125 \text{ J}$

$\boxed{K = 125 \text{ J}}$

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Masala 9: Impuls momenti

Massasi 0.2 kg, uzunligi 0.5 m bo'lgan tayoq uchi atrofida 4 rad/s bilan aylanmoqda. Impuls momentini toping.

Yechim

$I = \frac{1}{3}ML^2 = \frac{1}{3} \times 0.2 \times 0.5^2 = \frac{0.05}{3} = 0.0167$ kg·m²

$L = I\omega = 0.0167 \times 4 = 0.067 \text{ kg·m}^2/\text{s}$

$\boxed{L \approx 0.067 \text{ kg·m}^2/\text{s}}$

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Masala 10: Parallel o'qlar teoremasi

Massasi 4 kg, radiusi 0.1 m bo'lgan diskning chetidan o'tuvchi o'q uchun inertsiya momentini toping.

Yechim

Markazdan: $I_{cm} = \frac{1}{2}MR^2 = \frac{1}{2} \times 4 \times 0.01 = 0.02$ kg·m²

Parallel o'qlar: $I = I_{cm} + Md^2 = 0.02 + 4 \times 0.01 = 0.02 + 0.04 = 0.06$ kg·m²

$\boxed{I = 0.06 \text{ kg·m}^2}$

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O'rta (11-22)

Masala 11: Massa markazi

Massalari 2 kg va 3 kg bo'lgan nuqtalar mos ravishda $x = 0$ va $x = 5$ m da joylashgan. Massa markazini toping.

Yechim

$x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} = \frac{2 \times 0 + 3 \times 5}{2 + 3} = \frac{15}{5} = 3 \text{ m}$

$\boxed{x_{cm} = 3 \text{ m}}$

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Masala 12: Muvozanat

2 m uzunlikdagi taxtaning chap uchiga 50 N, o'ng uchiga 30 N yuk qo'yilgan. Tayanch qayerda bo'lishi kerak?

Yechim

Tayanch chap uchidan $x$ masofada bo'lsin.

Momentlar muvozanati (tayanch atrofida):

$50 \times x = 30 \times (2 - x)$

$50x = 60 - 30x$

$80x = 60$

$x = 0.75$ m

$\boxed{x = 0.75 \text{ m (chap uchidan)}}$

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Masala 13: Dumalanish tezligi

Disk balandligi 2 m bo'lgan nishablikdan sirpanmasdan dumalanib tushdi. Pastdagi tezligini toping.

Yechim

Disk uchun: $I_{cm} = \frac{1}{2}MR^2$, $\frac{I_{cm}}{MR^2} = 0.5$

$v = \sqrt{\frac{2gh}{1 + I/(MR^2)}} = \sqrt{\frac{2 \times 10 \times 2}{1 + 0.5}} = \sqrt{\frac{40}{1.5}} = \sqrt{26.67}$

$\boxed{v \approx 5.16 \text{ m/s}}$

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Masala 14: Shar vs Disk

Bir xil balandlikdan shar va disk dumalanib tushsa, qaysi biri tezroq?

Yechim

Shar: $I/(MR^2) = 2/5 = 0.4$

$v_{shar} = \sqrt{\frac{2gh}{1.4}} = \sqrt{1.43gh}$

Disk: $I/(MR^2) = 1/2 = 0.5$

$v_{disk} = \sqrt{\frac{2gh}{1.5}} = \sqrt{1.33gh}$

$v_{shar} > v_{disk}$

$\boxed{\text{Shar tezroq (chunki inertsiya koeffitsienti kichikroq)}}$

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Masala 15: Impuls momenti saqlanishi

Figurist qo'llarini yoyib $I_1 = 3$ kg·m², $\omega_1 = 2$ rad/s bilan aylanmoqda. Qo'llarini yig'ganda $I_2 = 1$ kg·m² bo'lsa, yangi burchak tezlik qancha?

Yechim

$L_1 = L_2$ (tashqi moment yo'q)

$I_1\omega_1 = I_2\omega_2$

$\omega_2 = \frac{I_1\omega_1}{I_2} = \frac{3 \times 2}{1} = 6$ rad/s

$\boxed{\omega_2 = 6 \text{ rad/s (3 marta tezroq)}}$

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Masala 16: Motor quvvati

Motor 1500 RPM da 20 N·m moment beradi. Quvvatini toping.

Yechim

$\omega = 1500 \times \frac{2\pi}{60} = 50\pi$ rad/s

$P = \tau\omega = 20 \times 50\pi = 1000\pi \approx 3142 \text{ W} \approx 3.14 \text{ kW}$

$\boxed{P \approx 3.14 \text{ kW}}$

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Masala 17: Robot qo'li momenti

Robot qo'li uzunligi 0.4 m, uchida 2 kg yuk. Qo'lni gorizontal tutish uchun kerakli momentni toping.

Yechim

$\tau = mgd = 2 \times 10 \times 0.4 = 8$ N·m

$\boxed{\tau = 8 \text{ N·m}}$

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Masala 18: Kompozit jism

Radiusi 0.2 m bo'lgan diskka (M = 2 kg) radiusi 0.1 m bo'lgan disk (m = 1 kg) markaziy ulangan. Umumiy inertsiya momentini toping.

Yechim

$I_1 = \frac{1}{2}M R_1^2 = \frac{1}{2} \times 2 \times 0.04 = 0.04$ kg·m²

$I_2 = \frac{1}{2}m R_2^2 = \frac{1}{2} \times 1 \times 0.01 = 0.005$ kg·m²

$I = I_1 + I_2 = 0.04 + 0.005 = 0.045$ kg·m²

$\boxed{I = 0.045 \text{ kg·m}^2}$

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Masala 19: Tezlashish vaqti

$I = 0.2$ kg·m² bo'lgan disk 5 N·m moment bilan aylantirilmoqda. 100 rad/s ga yetishi uchun qancha vaqt kerak?

Yechim

$\alpha = \frac{\tau}{I} = \frac{5}{0.2} = 25$ rad/s²

$\omega = \alpha t$

$t = \frac{\omega}{\alpha} = \frac{100}{25} = 4$ s

$\boxed{t = 4 \text{ s}}$

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Masala 20: Pretsessiya

Giroskop: $I = 0.01$ kg·m², $\omega = 100$ rad/s, massa markazi o'qdan 5 cm, M = 0.5 kg. Pretsessiya tezligini toping.

Yechim

$\tau = Mgd = 0.5 \times 10 \times 0.05 = 0.25$ N·m

$L = I\omega = 0.01 \times 100 = 1$ kg·m²/s

$\Omega_p = \frac{\tau}{L} = \frac{0.25}{1} = 0.25 \text{ rad/s}$

$\boxed{\Omega_p = 0.25 \text{ rad/s}}$

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Masala 21: Ag'darish burchagi

Balandligi 1 m, tayanch kengligi 0.4 m bo'lgan quti. Maksimal og'ish burchagini toping.

Yechim

$\theta_{max} = \arctan\left(\frac{b/2}{h}\right) = \arctan\left(\frac{0.2}{1}\right) = \arctan(0.2)$

$\boxed{\theta_{max} \approx 11.3°}$

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Masala 22: Ish-energiya

Disk tinch holatdan 50 N·m moment bilan 10 rad aylantirildi. Yakuniy burchak tezlikni toping ($I = 2$ kg·m²).

Yechim

$W = \tau\theta = 50 \times 10 = 500$ J

$W = \Delta K = \frac{1}{2}I\omega^2 - 0$

$\omega = \sqrt{\frac{2W}{I}} = \sqrt{\frac{2 \times 500}{2}} = \sqrt{500}$

$\boxed{\omega \approx 22.4 \text{ rad/s}}$

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Qiyin (23-30)

Masala 23: Quadcopter inertsiya

Quadcopter ramkasi: 4 ta motor (har biri 50 g) markazdan 15 cm masofada. Markaziy inertsiya momentini toping (ramka massasini hisobga olmang).

Yechim

Har bir motor nuqtaviy massa sifatida:

$I = 4 \times mr^2 = 4 \times 0.05 \times 0.15^2 = 4 \times 0.05 \times 0.0225$

$I = 4 \times 0.001125 = 0.0045$ kg·m²

$\boxed{I = 0.0045 \text{ kg·m}^2 = 45 \text{ g·cm}^2}$

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Masala 24: Propeller momenti

Propeller: $I = 5 \times 10^{-5}$ kg·m². 0 dan 10000 RPM gacha 0.2 s da yetishi uchun kerakli momentni toping.

Yechim

$\omega = 10000 \times \frac{2\pi}{60} = \frac{10000\pi}{30} = \frac{1000\pi}{3}$ rad/s

$\alpha = \frac{\omega}{t} = \frac{1000\pi/3}{0.2} = \frac{5000\pi}{3}$ rad/s²

$\tau = I\alpha = 5 \times 10^{-5} \times \frac{5000\pi}{3} = \frac{0.25\pi}{3}$

$\boxed{\tau \approx 0.26 \text{ N·m}}$

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Masala 25: Raketa spin stabilizatsiya

Raketa: $I = 0.5$ kg·m², spin tezligi 10 rad/s. 0.1 N·m tashqi moment qancha pretsessiya hosil qiladi?

Yechim

$L = I\omega = 0.5 \times 10 = 5$ kg·m²/s

$\Omega_p = \frac{\tau}{L} = \frac{0.1}{5} = 0.02 \text{ rad/s} = 1.15°/\text{s}$

$\boxed{\Omega_p = 0.02 \text{ rad/s} \approx 1.15°/\text{s}}$

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Masala 26: Yonilg'i sarfi va inertsiya

Raketa boshlang'ich massasi 10 kg, oxirgi 6 kg. Uzunligi 1 m (bir jinsli silindr). Inertsiya momenti qancha o'zgaradi?

Yechim

Silindr (bo'ylama o'q atrofida): $I = \frac{1}{2}MR^2$

Massaga proporsional, shuning uchun:

$\frac{I_f}{I_i} = \frac{M_f}{M_i} = \frac{6}{10} = 0.6$

$\boxed{I \text{ 40\% kamayadi}}$

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Masala 27: Yo-yo dinamikasi

Yo-yo: M = 0.1 kg, R = 0.03 m (disk). Ipdan qo'yib yuborilganda tezlanishni toping.

Yechim

Kuchlar: $Mg - T = Ma$ (chiziqli)

Momentlar: $TR = I\alpha = \frac{1}{2}MR^2 \cdot \frac{a}{R}$

$T = \frac{1}{2}Ma$

$Mg - \frac{1}{2}Ma = Ma$

$g = \frac{3}{2}a$

$a = \frac{2g}{3} = \frac{2 \times 10}{3} \approx 6.67 \text{ m/s}^2$

$\boxed{a = \frac{2g}{3} \approx 6.67 \text{ m/s}^2}$

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Masala 28: Ikki diskli sistema

Disk 1: $I_1 = 0.2$ kg·m², $\omega_1 = 10$ rad/s. Disk 2: $I_2 = 0.3$ kg·m², $\omega_2 = 0$. Birlashganda yakuniy tezlik?

Yechim

Impuls momenti saqlanishi:

$I_1\omega_1 + I_2\omega_2 = (I_1 + I_2)\omega_f$

$0.2 \times 10 + 0 = 0.5 \times \omega_f$

$\omega_f = \frac{2}{0.5} = 4$ rad/s

$\boxed{\omega_f = 4 \text{ rad/s}}$

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Masala 29: 3D inertsiya tensori

Quadcopter ramkasi (X konfiguratsiya): 4 ta motor (m = 50 g) koordinatalarda (0.1, 0.1), (-0.1, 0.1), (-0.1, -0.1), (0.1, -0.1) m. $I_{zz}$ ni toping.

Yechim

Z o'qi uchun: $I_{zz} = \sum m_i(x_i^2 + y_i^2)$

Har bir motor uchun: $r^2 = 0.1^2 + 0.1^2 = 0.02$ m²

$I_{zz} = 4 \times 0.05 \times 0.02 = 0.004$ kg·m²

$\boxed{I_{zz} = 0.004 \text{ kg·m}^2}$

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Masala 30: Dron yaw boshqaruvi

Quadcopter: $I_{zz} = 0.01$ kg·m². 90° burilish uchun 0.5 s vaqt berilgan. Kerakli momentni toping (tezlanish + sekinlashish bir xil).

Yechim

Yarim vaqt tezlanish, yarim sekinlashish.

$\theta = 90° = \frac{\pi}{2}$ rad

Tezlanish fazasi: $\theta_1 = \frac{1}{2}\alpha t_1^2 = \frac{1}{2}\alpha \times 0.25^2$

Sekinlashish fazasi: $\theta_2 = \omega_{max} \times 0.25 - \frac{1}{2}\alpha \times 0.25^2$

$\omega_{max} = \alpha \times 0.25$

Jami: $\theta = \frac{1}{2}\alpha \times 0.0625 + \alpha \times 0.0625 - \frac{1}{2}\alpha \times 0.0625 = \alpha \times 0.0625$

$\frac{\pi}{2} = \alpha \times 0.0625$

$\alpha = \frac{\pi}{2 \times 0.0625} = 8\pi$ rad/s²

$\tau = I\alpha = 0.01 \times 8\pi = 0.08\pi$

$\boxed{\tau \approx 0.25 \text{ N·m}}$

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Natijalar Jadvali

Daraja	Masalalar	Mavzular
Boshlang'ich	1-10	Burchak kinematika, I, $\tau$, K, L
O'rta	11-22	Massa markazi, dumalanish, muvozanat
Qiyin	23-30	Dron/raketa dinamikasi, 3D

Jami: 30 masala