System with external forces — conditions for conservation: Foundations

1.4.4 · D1 · Physics › Momentum & Collisions › System with external forces — conditions for conservation

Parent note ki master line $\frac{d P}{d t} = F_{e x t}^{n e t}$ pe trust karne se pehle, us mein har letter ka matlab aisa hona chahiye jo tum picture kar sako. Yeh page har symbol ko kuch bhi nahi se build karta hai, aise order mein jahan har ek sirf pehle walon pe rely karta hai.

1. Un quantities ke liye arrows jinki ek direction hoti hai:

Kyun zaroori hai? Kyunki 5 speed se left jaata hua ball aur 5 speed se right jaata hua ball alag physical situations hain, even though dono ka size 5 hai. Ek akela number unhe apart nahi bata sakta; ek arrow bata sakta hai.

Figure 1 mein red arrow ki length size hai (kitna) aur uska tilt direction hai (kis taraf). Bas yahi ek vector hai — length ke saath direction.

Figure 2 teeno cheezein dikhata hai: tip-to-tail sum, ek arrow aur uska opposite $- b$ , aur woh kaise zero-length dot $0$ mein collapse hote hain.

2. Mass: $m$ aur $M$

Picture karo: bricks se bhari shopping trolley ( $M$ large) ek khaali ke muqable ( $m$ small). Wohi push khaali wali ko bahut tezi se move karta hai. Topic ko $m$ chahiye kyunki momentum mass times velocity hai — "kitna stuff" is ki kahaani ka aadha hissa hai.

3. Velocity: $v$

Vector kyun, sirf speed kyun nahi? Kyunki ek collision mein do objects ki equal speed $v$ ho sakti hai lekin head-on-opposite velocities $v$ , aur poora outcome isi pe depend karta hai. Speed akela bilkul wahi information throw away kar deta jo humein chahiye.

4. Momentum: $p$ aur total $P$

Figure 3 ek chhoti halki object aur ek badi bhaari object dikhata hai jinka momentum arrow same hai: bhaari wali dheerey chalti hai lekin zyada mass hai, aur arrows match karte hain. Yeh balance hai jo $p = m v$ visible banata hai.

Subscript $i$ ek counter hai: $m_{i} v_{i}$ ka matlab hai " $i$ -th object ka mass aur velocity." Toh $\sum_{i} m_{i} v_{i} = m_{1} v_{1} + m_{2} v_{2} + \dots$ . Yahi poora notation hai — kuch aur nahi.

5. Force: $F$ (aur internal / external split)

Double subscript $f_{ij}$ padhte hain " $i$ pe, $j$ se force." Toh $f_{12}$ object 2 se object 1 pe force hai, aur $f_{21}$ uska twin hai, object 1 se object 2 pe.

Figure 4 mein dashed loop system boundary hai — woh line jo tum draw karte ho decide karne ke liye ki kaun "andar" hai. Loop cross karne wale arrows external hain (red); poori tarah andar rehne wale internal hain (black pairs). Is topic mein sab kuch isi pe hinge karta hai ki tum woh loop kahan draw karte ho.

Intuition Internal pairs

$0$ mein kyun add hote hain (sketch) Koi bhi pair lo, members $i$ aur $j$ . $j$ se $i$ pe push $f_{ij}$ hai, aur $i$ se $j$ pe push $f_{j i}$ hai. Yeh do size mein equal lekin direction mein opposite hain — apart push karte hue ya saath kheenchte hue, hamesha symmetric (yeh equal-and-opposite twinning Newton's Third Law hai). Section 1 ki language mein, yeh bilkul $f_{j i} = - f_{ij}$ hai, toh twin kuch bhi nahi add karta: $f_{ij} + f_{j i} = f_{ij} + (- f_{ij}) = 0 .$ Har internal force aise pair ka ek aadha hissa hai, aur jab tum poore system ke internal forces add karte ho toh woh pair by pair cancel hote hain: $\sum_{i} \sum_{j \neq = i} f_{ij} = 0 .$ Yahi wajah hai ki internal pushes — chahe kitne bhi violent hon, jaise ek explosion — total $P$ ko kabhi nahi hilate: unke arrows hamesha ek equal-and-opposite twin ke saath aate hain jo unhe erase kar deta hai.

6. Rate of change: derivative $\frac{d}{d t}$

Yeh woh tool hai jisse parent note shuru karta hai. Kuch aur assume nahi kiya gaya — reverse view ke liye Impulse–Momentum Theorem dekho (force ko time ke upar add karke total change pana).

7. Sab kuch jod ke: master equation padhna

Ab $\frac{d P}{d t} = F_{e x t}^{n e t}$ ka har symbol earn kiya hua hai: $P$ (Section 4) group ka total momentum arrow hai; $\frac{d}{d t}$ (Section 6) poochh raha hai yeh kitna fast change hota hai; aur $F_{e x t}^{n e t} = \sum_{i} F_{i}^{e x t}$ (Section 5) summed-up bahari push hai, internal ones ke pairs mein cancel hone ke baad. Poori line kehti hai: "group ka total momentum utni hi fast change hota hai jitna bahari push karta hai." Agar woh net bahari push $0$ hai, toh total bilkul nahi badlta — yahi conservation hai.

Recall Notation par quick self-check

$p$ pe arrow tumhe kya batata hai jo plain $p$ nahi batata? ::: Uska direction, sirf size nahi. $∣ v ∣$ (ya plain $v$ ) ka matlab kya hai, aur iska everyday naam? ::: Velocity arrow ki length jisme direction discard kar di gayi — uski speed. $f_{ij}$ mein kaun sa object force feel karta hai? ::: Object $i$ (pehla subscript) feel karta hai, object $j$ se aake. $\sum_{i}$ tumhe kya karne ki instruction deta hai? ::: System mein har object ke liye ek term add karo. $F_{e x t}^{n e t}$ ko sum ke roop mein likho. ::: $F_{e x t}^{n e t} = \sum_{i} F_{i}^{e x t}$ — har member pe har bahari push, arrows ki tarah add karke. $\sum_{i} \sum_{j \neq = i} f_{ij} = 0$ kyun hai? ::: Internal forces equal-and-opposite twins $f_{j i} = - f_{ij}$ ke roop mein aate hain, jo pair by pair cancel hote hain.

Prerequisite map

Neeche diagram is page pe use ki gayi build order trace karta hai: ise bottom-up padhna. Arrows (Section 1) aur mass (Section 2) velocity aur phir momentum mein combine hote hain (Sections 3–4); woh total $P$ mein add ho jaate hain. Alag se, force split (Section 5) aur derivative (Section 6) Newton's 2nd law dete hain. Total momentum, internal/external split, aur Newton's 2nd law sab master equation ko feed karte hain, jo wahi conservation wala parent topic hai. (Agar diagram tumhare reader mein render na ho, toh wohi dependency order is paragraph mein spell out ki gayi hai.)

Yeh bhi dekho Newton's Third Law (internal pairs kyun cancel hote hain) aur Center of Mass Motion (wohi master equation dobara likhi hui).

Equipment checklist

Right side cover karo aur dekho ki reveal karne se pehle har ek state kar sako.

Vector hota hai... ::: ek quantity jisme size (length) aur direction (tilt) dono hote hain, arrow ke saath likha jaata hai.
$v$ ka size, $∣ v ∣$ ya $v$ likha hua, hota hai... ::: direction discard karke iski length; velocity ke liye ise speed kehte hain.
Do arrows add karne ka matlab hai... ::: unhe tip-to-tail rakhna; start se finish tak ka arrow sum hota hai.
$b$ subtract karne ka matlab hai... ::: $- b$ add karna, same-length arrow opposite direction mein; aur $b + (- b) = 0$ .
Zero vector $0$ hota hai... ::: koi length nahi wala arrow (ek dot) — jo ek equal-and-opposite pair add up karke banta hai.
Mass $m$ measure karta hai... ::: stuff ki miqdar — ek object ko speed up ya stop karna kitna mushkil hai.
Ek object ka momentum hota hai... ::: $p = m v$ , velocity ke saath ek arrow, zyada mass ke liye lamba.
Total momentum $P$ hota hai... ::: $\sum_{i} m_{i} v_{i}$ , saare chhote momentum arrows tip-to-tail add kiye hue.
Internal force hoti hai... ::: chosen system ke do members ke beech push, $f_{ij}$ likha jaata hai.
External force hoti hai... ::: system ke bahar se ek system member pe push, $F^{e x t}$ .
Net external force $F_{e x t}^{n e t}$ hoti hai... ::: $\sum_{i} F_{i}^{e x t}$ , har member pe har bahari push arrows ki tarah add ki hui.
Internal forces kyun cancel hoti hain... ::: woh twins $f_{j i} = - f_{ij}$ mein aate hain, toh $\sum_{i} \sum_{j \neq = i} f_{ij} = 0$ .
$\frac{d p}{d t}$ ka matlab hai... ::: momentum abhi kitni fast change ho raha hai (per instant rate of change).
Newton's 2nd law momentum form mein hai... ::: $F = \frac{d p}{d t}$ — force woh hai jo momentum ko time ke saath change karti hai.

Ready ho? Toh parent topic pe jao aur master equation plain English ki tarah padhegi.