2.3 Energiya va Impuls — Nazariya

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Kirish

Energiya va impuls — fizikaning eng fundamental tushunchalari. Ular saqlanish qonunlariga bo'ysunadi, bu esa murakkab tizimlarni tahlil qilishni osonlashtiradi.

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1. Ish (Work)

Ta'rif

Kuch jismni siljitganda bajarilgan ish:

$$W = \vec{F} \cdot \vec{d} = Fd\cos\theta$$

• $\theta$ — kuch va siljish orasidagi burchak
• Birlik: Joul (J) = N·m = kg·m²/s²

Maxsus holatlar

Holat	Burchak	Ish
Kuch va harakat bir yo'nalishda	0°	$W = Fd$ (ijobiy)
Kuch harakatga perpendikular	90°	$W = 0$
Kuch harakatga qarshi	180°	$W = -Fd$ (manfiy)

O'zgaruvchan kuch

$$W = \int_{x_1}^{x_2} F(x) \, dx$$

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2. Kinetik Energiya

Harakatdagi jismning energiyasi:

$$KE = \frac{1}{2}mv^2$$

Ish-Energiya Teoremasi

$$W_{net} = \Delta KE = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2$$

Natijaviy ish = kinetik energiya o'zgarishi

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3. Potensial Energiya

Gravitatsion potensial energiya

$$PE_g = mgh$$

$h$ — tanlangan "nol" sathdan balandlik.

Elastik potensial energiya (prujina)

$$PE_s = \frac{1}{2}kx^2$$

$x$ — muvozanat holatidan siljish.

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4. Mexanik Energiya Saqlanishi

Agar faqat konservativ kuchlar (gravitatsiya, prujina) ish qilsa:

$$E_{total} = KE + PE = \text{const}$$ $$\frac{1}{2}mv_1^2 + mgh_1 = \frac{1}{2}mv_2^2 + mgh_2$$

Konservativ va nokonservativ kuchlar

Konservativ	Nokonservativ
Gravitatsiya	Ishqalanish
Prujina	Havo qarshiligi
Elektrostatik	Motor kuchi

Nokonservativ kuchlar mavjud bo'lganda:

$$E_1 + W_{nc} = E_2$$

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5. Quvvat (Power)

Ishning bajarilish tezligi:

$$P = \frac{dW}{dt} = \frac{W}{\Delta t}$$

Yoki:

$$P = \vec{F} \cdot \vec{v} = Fv\cos\theta$$

• Birlik: Vatt (W) = J/s

Foydali formulalar

$$1 \text{ hp (ot kuchi)} = 746 \text{ W}$$ $$1 \text{ kWh} = 3.6 \times 10^6 \text{ J}$$

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6. Impuls (Momentum)

Ta'rif

$$\vec{p} = m\vec{v}$$

• Birlik: kg·m/s

Nyuton II qonuni (umumiy shakl)

$$\vec{F} = \frac{d\vec{p}}{dt}$$

Agar massa doimiy: $F = ma$

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7. Impuls (Impulse)

Kuch × vaqt:

$$\vec{J} = \vec{F} \cdot \Delta t = \Delta \vec{p}$$

O'zgaruvchan kuch uchun:

$$\vec{J} = \int_{t_1}^{t_2} \vec{F} \, dt$$

Amaliy qo'llanish

Raketada: $J = F \cdot t$ = dvigatel impulsi

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8. Impuls Saqlanish Qonuni

Yopiq sistemada (tashqi kuchlar yo'q):

$$\vec{p}_{total} = \text{const}$$ $$m_1\vec{v}_{1i} + m_2\vec{v}_{2i} = m_1\vec{v}_{1f} + m_2\vec{v}_{2f}$$

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9. To'qnashuvlar

Elastik to'qnashuv

• Impuls saqlanadi ✓
• Kinetik energiya saqlanadi ✓

Bir o'lchovda:

$$v_{1f} = \frac{(m_1-m_2)v_{1i} + 2m_2 v_{2i}}{m_1+m_2}$$ $$v_{2f} = \frac{(m_2-m_1)v_{2i} + 2m_1 v_{1i}}{m_1+m_2}$$

Noelastik to'qnashuv

• Impuls saqlanadi ✓
• Kinetik energiya saqlanmaydi ✗

To'liq noelastik (jismlar birlashadi):

$$(m_1 + m_2)v_f = m_1 v_{1i} + m_2 v_{2i}$$

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10. Burchak Impulsi (Angular Momentum)

$$\vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}$$

Saqlanish qonuni

Tashqi moment yo'q bo'lsa:

$$\vec{L} = I\omega = \text{const}$$

Masalan, figure skater qo'llarini yig'ganda $I$ kamayadi, $\omega$ ortadi.

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11. Robotikada Qo'llanilish

Motor tanlash

Kerakli quvvat:

$$P = Fv = \frac{\tau \cdot \omega}{\eta}$$

$\eta$ — samaradorlik (efficiency)

Raketa tenglamasi

Tsiolkovsky:

$$\Delta v = v_e \ln\frac{m_0}{m_f}$$

• $v_e$ — exhaust tezligi
• $m_0/m_f$ — massa nisbati

Dron energiya sarfi

Hovering:

$$P = \frac{(mg)^{3/2}}{\sqrt{2\rho A}}$$

Uchish vaqti:

$$t = \frac{E_{battery}}{P}$$

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Xulosa

Tushuncha	Formula	Saqlanish
Kinetik energiya	$KE = \frac{1}{2}mv^2$	Elastik to'qnashuvda
Potensial energiya	$PE = mgh$	—
Impuls	$p = mv$	Yopiq sistemada doim
Quvvat	$P = Fv$	—

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