Rigid body — definition, degrees of freedom: Foundations

1.5.1 · D1 · Physics › Rotational Mechanics › Rigid body — definition, degrees of freedom

Kuch bhi count karne se pehle, humein agree karna hoga ki parent page par har symbol ka matlab actually kya hai — ek picture ki tarah, na ki ek squiggle ki tarah. Hum unhe order mein build karte hain, taki koi bhi cheez use hone se pehle earn na ho.

1. Space mein ek point — aur numbers $(x, y, z)$

Apne room ke ek corner ko "zero" maan lo. Ek arrow right, ek forward, ek up — teeno arrows right angles par. Room mein dhool ke kisi bhi kaad tak pahunchne ke liye tum kuch right, kuch forward, kuch up chalte ho. Woh teen amounts hain $x$ , $y$ , $z$ .

Figure — Rigid body — definition, degrees of freedom

Yeh topic ko kyun chahiye: parent ka "3 DOF per particle" literally yahi teen numbers hain. Baaki sab kuch is idea ki copies hain jisme rules subtract kiye gaye hain.

2. Position vector $r$ — teen numbers ki jagah ek arrow

"3 right, 2 forward, 5 up" kehne ki jagah, hum corner se seedha dhool ke kaad tak ek arrow draw karte hain. Woh arrow hi $r$ hai. Iske andar secretly teen numbers $(x, y, z)$ abhi bhi hain — woh Section 1 ke teen arrows ke along uske shadow-lengths hain — lekin unhe ek symbol mein bundle karne se equations chhoti rehti hain.

$r_{i}$ mein subscript $i$ ka matlab sirf "particle number $i$ ke liye arrow" hai — particle 1 ka $r_{1}$ hai, particle 2 ka $r_{2}$ hai, aur aise hi aage. Yeh ek name tag hai, kuch nahi.

Yeh topic ko kyun chahiye: rigidity rule $r_{i}$ aur $r_{j}$ ke saath likha gaya hai. Hume pata hona chahiye ki yeh sirf particles ko labelled arrows hain isse pehle ki woh rule kuch matlab rakhe.

3. Do arrows ka difference $r_{i} - r_{j}$ aur uski length $∣ \cdot ∣$

Yahan parent ki definition ka core hai, ek piece at a time build kiya gaya hai.

Figure dekho: origin se do arrows nikalte hain, ek $i$ tak, ek $j$ tak. Unke tips ko join karne wala green arrow $r_{i} - r_{j}$ hai. Arrows ka subtraction matlab hai "tip minus tip," aur geometrically yeh ek particle se doosre tak ka bridge hai.

Toh $∣ r_{i} - r_{j} ∣$ particle $i$ aur particle $j$ ke beech ki straight-line distance hai. Bas itna hi kehta hai woh scary symbol.

4. Counting notation: $N$ , $3 N$ , aur "constraints"

5. Choosing symbol $(2 N)$

$N = 4$ ke liye: $(2 4) = \frac{4 \cdot 3}{2} = 6$ pairs. Har pair ka ek fixed-distance rule hai — lekin parent warn karta hai ki $N$ badhne par yeh rules sab independent nahi hote. Isliye hum blindly pairs count karne ki jagah 3 points fix karte hain.

Figure $3 + 2 + 1 = 6$ construction dikhata hai: particle 1 free (3), particle 2 uske around ek sphere par (2 bacha), particle 3 ek circle par (1 bacha). Teen points freeze karne se poori body freeze ho jaati hai.

Reference point jiska $(x, y, z)$ hum track karte hain woh usually Centre of Mass hota hai.
Jab body spin karti hai, spin karna kitna mushkil hai woh Moment of Inertia hai, aur time ke saath angle $θ$ kaise badalta hai woh Rotational Kinematics hai.
Constraints subtract karke DOF count karna exactly Constraints and Lagrangian Mechanics ki language hai.
"Diatomic ke liye 5 DOF" fact gases ke Equipartition Theorem ko power karta hai.
Ek rigid body jo slide aur spin dono kare ek matching condition ke saath woh Rolling Motion kar rahi hai.

Yeh sab parent Rigid body — definition, degrees of freedom ke under rehte hain.

Equipment checklist

Right side cover karo aur aage badhne se pehle har ek ka jawab do.

(x, y, z)

teen numbers kya measure karte hain?

Origin se kitni door ek point right, forward, aur up directions mein baitha hai.

r_{i}

ek picture mein kya hai?

Origin se particle

i

tak ek arrow; yeh us particle ke teen coordinates bundle karta hai.

r_{i} - r_{j}

kaun sa arrow draw karta hai?

Particle

j

se particle

i

tak ka arrow (tip minus tip).

∣ r_{i} - r_{j} ∣

mein bars kya dete hain?

Do particles ke beech ki straight-line distance — ek plain non-negative number.

Rigidity condition words mein batao.

Har pair-distance time mein constant rehti hai, isliye shape kabhi nahi badalti.

Constraint kya hai, aur ek count ke saath kya karta hai?

Coordinates ko link karne wali ek equation; har independent constraint exactly ek free number hata deta hai.

(2 4)

compute karo aur batao yeh kya count karta hai.

6

— 4 particles mein distinct particle pairs ki number.

Hum

(2 N)

ko hamesha constraint count kyun trust nahi kar sakte?

N > 4

ke liye pair-distance equations redundant hoti hain; sirf

3 N - 6

independent hote hain.

Hinge par door ke liye

θ

kya single number supply karta hai?

Swing angle — uska ek remaining degree of freedom.

3D mein kitne independent rotation axes hote hain?

Teen (roll, pitch, yaw), jo 3 rotational DOF dete hain.

Master DOF formula likho.

DOF = 3 N - (independent constraints)

Foundations — Rigid body — definition, degrees of freedom

1. Space mein ek point — aur numbers $(x, y, z)$

2. Position vector $r$ — teen numbers ki jagah ek arrow

3. Do arrows ka difference $r_{i} - r_{j}$ aur uski length $∣ \cdot ∣$

4. Counting notation: $N$ , $3 N$ , aur "constraints"

5. Choosing symbol $(2 N)$

6. Angle symbol $θ$ aur "rotation axis"

Yeh foundations topic ko kaise feed karte hain

Yeh tools aage kahan jaate hain

Equipment checklist