Foundations — Motion of centre of mass — external force determines a_CM

1.4.11 · D1 · Physics › Momentum & Collisions › Motion of centre of mass — external force determines a_CM

${\vec{F}}_{ext}=M{\vec{a}}_{CM}$ padhne se pehle, tumhe isme har ek mark padhna aana chahiye. Yeh page har symbol ko kuch nahi se banata hai, us order mein jisme woh ek doosre par depend karte hain. Agar koi mark parent page par dikhe, toh woh pehle yahan define hoga.

0 — Upar arrow wale symbol ka matlab

Parent page ki sabse pehli assumption hai yeh chhota sa arrow: $\vec{r}$, $\vec{v}$, $\vec{F}$.

Picture. Ek chosen origin point $O$ par khade ho jao. Kisi cheez ki location describe karne ke liye sirf "5 metres" nahi kehte — kehte ho "5 metres, us taraf." Yeh ek arrow hai $O$ se object tak.

Topic ko yeh kyun chahiye. Positions, velocities, aur forces sabke paas directions hain. Ek skater left push kar raha hai yeh right push karne se bilkul alag hai. Ek plain number (jise scalar kehte hain, koi arrow nahi) kabhi left aur right ka fark nahi bata sakta. Isliye parent page par har core symbol par arrow hai.

1 — $m_i$ : har piece ki mass, aur index $i$

Picture. Numbered marbles imagine karo: marble 1, marble 2, marble 3. Har ek ki apni mass hai. Index $i$ woh sticker hai jo hum har marble par lagate hain taaki hum naye letters ke bina unke baare mein ek at a time baat kar sakein.

Topic ko yeh kyun chahiye. System mein bahut saare particles hain. Hume ek flexible symbol chahiye, $m_i$, jo "unme se koi bhi" ke liye khade ho, taaki ek hi formula 2 ya 2000 pieces ke system ko cover kare.

2 — ${\sum }_i$ : summation sign (sabhi tags par add karo)

Picture. Ek basket jisme tum har particle ke liye ek term daalo, phir total padho.

Topic ko yeh kyun chahiye. $M={\sum }_i{m}_i$ (total mass) aur ${\vec{R}}_{CM}=\frac{1}{M}{\sum }_i{m}_i{\vec{r}}_i$ (balance point) dono hain "har particle ke liye yeh karo aur add karo." $\sum$ ke bina hume har naye number of particles ke liye formula rewrite karna padta.

3 — ${\vec{r}}_i$ : position, aur ${\vec{R}}_{CM}$ : balance point

Ab hum show ke star ko build kar sakte hain.

Picture — mass se weight kyun. Ek light ball aur ek heavy ball ko seesaw par rakho. Balance point midpoint nahi hai; woh heavy ball ke paas baithta hai. Mass-weighting bilkul yahi bake in karta hai.

Topic ko yeh kyun chahiye. Yahi ek point hai jo poora topic track karta hai. "Internal forces CM ko move nahi kar sakti" ka claim is specific average ke behave karne ke baare mein ek claim hai.

4 — ${\vec{v}}_i$, ${\vec{a}}_i$ : velocity aur acceleration (rates of change)

Parent page position formula ko "differentiate" karta hai. Yahan woh word ka matlab hai, bina kisi calculus jargon ke.

Tool kyun, aur koi doosra kyun nahi? Hum jaanna chahte hain ki balance point kaise move karta hai aur speed up karta hai. Movement position ka time ke saath change hai; rate of change precisely woh tool hai jo "per unit time" measure karta hai. Isliye parent do baar rates of change leta hai — ek baar velocity tak pahunchne ke liye, ek baar phir acceleration tak.

5 — ${\vec{p}}_i$ aur ${\vec{P}}_{total}$ : momentum

Picture. Ek slow truck aur ek fast bike same momentum carry kar sakte hain; rest par truck koi momentum carry nahi karta. Arrow ki length oomph hai, uski direction travel direction hai.

Topic ko yeh kyun chahiye. Gem ${\vec{P}}_{total}=M{\vec{V}}_{CM}$ (parent Step 2) bheed ke total oomph ko single balance point ki motion se link karta hai — conservation of momentum ka bridge.

6 — ${\vec{F}}_i$, aur internal/external split

Topic ko yeh kyun chahiye. Poora result in dono ko alag karne par depend karta hai. Internal forces cancel ho jayengi; external forces CM ko drive karne ke liye bachegi.

7 — Newton's Third Law : kyun internal pile cancel hoti hai

Picture. Do skaters dhakelte hain: dono ko ek doosre se door push lagti hai, size mein equal, direction mein opposite. Pair add karo toh zero arrow milta hai — woh annihilate ho jaate hain.

Topic ko yeh kyun chahiye. ${\sum }_i{\sum }_{j\ne i}{\vec{f}}_{ij}$ mein, har internal force apne equal-and-opposite twin ke saath appear hoti hai. Sum karne par, har twin-pair hai ${\vec{f}}_{ij}+{\vec{f}}_{ji}=\vec{0}$. Poori internal pile vanish ho jaati hai — chahe forces kitni bhi badi hon — sirf external forces bachti hain. Dekho Newton's Third Law.

8 — $M$, ${\vec{V}}_{CM}$, ${\vec{a}}_{CM}$ : whole-system symbols

Topic ko yeh kyun chahiye. Yahi symbols hain master result ${\vec{F}}_{ext}=M{\vec{a}}_{CM}$ aur $\vec{P}=M{\vec{V}}_{CM}$ mein. Ab un boxed formulas mein har mark ka ek plain-words meaning aur peeche ek picture hai.

Yeh foundations topic ko kaise feed karte hain

Equipment checklist

Khud test karo — right side cover karo, jawab do, phir reveal karo.

Connections

• Parent topic (Hinglish) — woh page jin foundations ko yeh unlock karta hai.
• Newton's Third Law — woh reason kyun internal forces pairs mein cancel hoti hain.
• Conservation of Linear Momentum — $\vec{P}=M{\vec{V}}_{CM}$ kya ban jaata hai jab ${\vec{F}}_{ext}=0$.
• Collisions (Elastic & Inelastic) — momentum aur CM ideas in action.
• Rocket Propulsion — external thrust versus internal mass ejection.
• Rigid Body Dynamics — CM as the point where $\vec{F}=M\vec{a}$ extended bodies ke liye hold karta hai.