Newton's second law — F = ma (net force), impulse-momentum form

1.2.2 · Physics › Newton's Laws & Dynamics

Newton's Second Law HAI KYA?

"Net" kyun? Kyunki sirf total push/pull matter karta hai. Agar tum ek box ko right mein 10 N se khincho aur friction left mein 10 N se khinche, toh net force zero hai aur $a = 0$ — chahe forces clearly exist kar rahe hon. Forces vectors hain; unhe tip-to-tail add karo.

$F = ma$ momentum form se KAISE milta hai? (Derivation from scratch)

Fundamental statement se shuru karo: $F_{net} = \frac{d p}{d t} = \frac{d ( m v )}{d t}$

Product rule apply karo (kyunki dono $m$ aur $v$ change ho sakte hain): $\frac{d ( m v )}{d t} = m \frac{d v}{d t} + v \frac{d m}{d t}$

Yeh step kyun? Momentum ek product $m v$ hai; calculus kehta hai product ki derivative ke do pieces hote hain.

Ab assume karo ki mass constant hai (ek single rigid object, koi leaking rocket nahi), toh $\frac{d m}{d t} = 0$ : $F_{net} = m \frac{d v}{d t} = m a$

Yeh step kyun? $\frac{d v}{d t}$ acceleration ki definition hi hai. Ho gaya.

Impulse–Momentum Theorem KAISE milta hai? (Derivation)

Fundamental form lo aur dono sides ko $d t$ se multiply karo, phir time interval $t_{1} \to t_{2}$ par integrate karo: $\int_{t_{1}}^{t_{2}} F_{net} d t = \int_{t_{1}}^{t_{2}} \frac{d p}{d t} d t = p_{2} - p_{1}$

Yeh step kyun? Time par ek derivative ko integrate karne se sirf net change milta hai (Fundamental Theorem of Calculus).

Left side ko impulse $J$ define karo:

Agar force constant hai, toh integral ek simple multiplication ban jaata hai: $J = F Δ t = Δ p$

Figure — Newton's second law — F = ma (net force), impulse-momentum form

Worked Examples

Common Mistakes (Steel-manned)

Recall Feynman: 12-saal ke bachche ko explain karo

Socho tum ek shopping trolley dhakka de rahe ho. Zyada dhakka → zyada tezi se speed badhti hai. Zyada bhaari → speed badhana mushkil. Yahi hai $F = ma$ : dhakka (force) = bhaari-pan (mass) × kitni tezi se speed change hoti hai (acceleration). Ab "impulse" bas yeh hai: sirf kitna zor lagao yeh matter nahi karta, balki kitni der tak bhi matter karta hai. Dheema dhakka lambe time tak wahi kaam kar sakta hai jo ek dum hard shove thodi der ke liye kare. Isliye fast ball pakdte waqt haath peeche kheench lete ho — time badhate ho taaki dard (force) kam ho.

Flashcards

Newton's 2nd law ka sabse fundamental statement kya hai?

F_{net} = \frac{d p}{d t}

— net force, momentum ke change ki rate ke barabar hai.

F = ma

kab fail hota hai, aur uski jagah kya use hota hai?

Jab mass change ho; use karo

F = \frac{d ( m v )}{d t} = m a + v \frac{d m}{d t}

Momentum form se

F = ma

derive karo.

F = \frac{d ( m v )}{d t} = m \frac{d v}{d t} + v \frac{d m}{d t}

; constant mass ⇒ doosra term 0 ⇒

F = m a

Impulse–momentum theorem batao.

J = \int F d t = Δ p = m v_{f} - m v_{i}

Impulse ki units?

N·s, jo kg·m/s ke barabar hai (momentum ke same).

Constant force ke liye, impulse = ?

J = F Δ t

Airbags injury kyun reduce karte hain?

Yeh stopping time

Δ t

badhate hain, toh same

Δ p

ke liye force

F = Δ p /Δ t

kam ho jaati hai.

Saare objects mass ki parwah kiye bina

g

acceleration se kyun girte hain?

a = F / m = m g / m = g

; mass cancel ho jaata hai.

Ek 2 kg block: 10 N right, 6 N friction left. Acceleration?

Net 4 N →

a = 2 m/s^{2}

right.

F_{n e t} = ma

mein "net" ka matlab kya hai?

Object par lagte saare forces ka vector sum.

Connections

Newton's First Law (Inertia) — Second Law ka special case $F_{n e t} = 0$ .
Newton's Third Law — forces pairs mein aate hain; wahi $F$ supply karta hai jo tum plug in karte ho.
Conservation of Linear Momentum — follow karta hai jab $F_{n e t} = 0$ .
Collisions and Elasticity — impulse–momentum contact par apply hota hai.
Friction — net force mein ek common contributor.
Variable Mass Systems & Rocket Equation — jahan sirf $\frac{d p}{d t}$ kaam karta hai.
Work-Energy Theorem — force ka space-integral (impulse ke opposite, jo time-integral hai).