Rigid body — definition, degrees of freedom

1.5.1 · Physics › Rotational Mechanics

Rigid body kya hota hai?

YEH definition kyun? "Rigid" ka poora matlab hi no deformation hai. "Shape kabhi nahi badlega" — yeh kehne ka ek hi mathematical tarika hai: saari inter-particle distances ko freeze kar do. Agar distances lock hain, toh object phir bhi move (translate) aur turn (rotate) kar sakta hai, lekin kabhi squash, stretch, ya twist nahi ho sakta.

Degrees of freedom (DOF)

"Independent" ka matlab kya hai? Ek coordinate independent hai agar tum use baaki coordinates se constraints ki madad se compute nahi kar sakte. Constraints woh equations hain jo coordinates ko baandhti hain (jaise fixed-distance rule).

Rigid body ke DOF derive karna — scratch se

Hum count karenge, phir constraints subtract karenge. Yeh master method hai: $DOF = (total coordinates) - (independent constraint equations)$

Step 1 — Single free particle

3D space mein ek point ko $(x, y, z)$ chahiye. Toh 3 DOF per free particle. Kyun? Teen perpendicular directions; har ek ke along ek number chahiye.

Step 2 — $N$ free particles (no constraints)

$coordinates = 3 N$ Kyun? Har particle ko independently 3 numbers chahiye.

Step 3 — Unhe rigid body mein lock karo

Ab impose karo: har pair-distance fixed hai. $N$ particles mein pairs ki sankhya hai $(2 N) = \frac{N ( N - 1 )}{2}$ Naively yahi constraints ki sankhya lagti hai. Lekin yeh constraints sab independent NAHI hain. Jab ek baar kuch points nail ho jaate hain, baaki over-determined ho jaate hain.

Step 4 — Smart trick: 3 points fix karo, baaki automatic hain

Poori rigid body locate karne ke liye, sirf 3 non-collinear points ki positions fix karna kaafi hai (socho: ek tabletop ke teen corners pin karo — poori table tab freeze ho jaati hai).

Point 1: free, 3 coordinates.
Point 2: Point 1 se fixed distance par constrained → 1 DOF lose → 2 coordinates (yeh Point 1 ke around ek sphere par rehta hai).
Point 3: Point 1 aur Point 2 dono se fixed distance → 2 DOF lose → 1 coordinate (yeh ek circle par rehta hai).

$DOF = 3 + 2 + 1 = 6$

Har doosra particle ( $4, 5, \dots, N$ ) in 3 points se apni fixed distances ke zariye poori tarah determined hota hai — 0 new DOF each.

Figure — Rigid body — definition, degrees of freedom

Special cases (hamesha pooche jaate hain!)

System	DOF	Kyun
Single particle (3D)	3	$(x, y, z)$
Rigid body in 3D	6	3 trans + 3 rot
Rigid body in 2D (plane)	3	$(x, y)$ + 1 rotation angle $θ$
Rigid body fixed at a point	3	sirf rotation (3 rot)
Rigid body on a fixed axle (1 axis fixed)	1	sirf spin angle $θ$
Diatomic molecule (2 point atoms)	5	3 trans + 2 rot (bond axis ke around spin point masses ke liye count nahi hota)

Common mistakes

Flashcards

Rigid body define karo.

Particles ka ek aisa system jahan har do ke beech ki doori time ke saath constant rehti hai (no deformation).

Rigidity ki mathematical condition?

∣ r_{i} - r_{j} ∣ =

constant for all pairs

(i, j)

Degrees of freedom kya hote hain?

Independent coordinates ki woh sankhya jo system ki configuration completely specify karne ke liye chahiye.

DOF ka master formula?

DOF = (total coordinates) − (number of independent constraints).

3D mein ek free particle ke DOF?

3D mein free rigid body ke DOF?

6 (3 translational + 3 rotational).

Rigid body ka DOF N se independent kyun hai?

3 non-collinear points fix karne ke baad (3+2+1=6), har doosre particle ki position fixed distances se forced hoti hai, 0 DOF add hota hai.

Plane mein confined rigid body ke DOF?

3 —

(x, y)

plus ek rotation angle.

Ek point par fixed rigid body ke DOF?

3 (sirf rotation).

Fixed axis wali rigid body ke DOF (jaise wheel on axle)?

1 (spin angle).

Diatomic (point-mass) molecule ke DOF?

5 — 3 translation + 2 rotation (bond axis ke around spin 0 count hota hai).

N rigid particles ke liye, kitne independent constraints hote hain?

3 N - 6

Recall Feynman: ek 12-saal ke bacche ko explain karo

Socho ek LEGO castle jo poori tarah glue se chipki hai taaki koi piece doosre ke relative move na kar sake. Apne dost ko exactly batane ke liye ki castle kahaan hai aur kis taraf face kar rahi hai, tum kitni cheezein bataoge? Pehle, kahaan rakhi hai — left/right, forward/back, up/down: 3 numbers. Phir kis taraf face kar rahi hai — ek airplane ki tarah teen taraf tilt karo (nose up/down, roll, turn): 3 aur. Total 6 numbers, aur koi farak nahi padta castle mein 10 bricks hain ya 10,000 — kyunki sab glued hain, ek baar 6 numbers place karo aur poori cheez lock ho jaati hai. Woh "6" hi hai jo hum degrees of freedom kehte hain.

Connections

Centre of Mass — 3 translational DOF ke liye natural reference point.
Moment of Inertia — 3 rotational DOF govern karta hai.
Rotational Kinematics — describe karta hai ki rotational DOF time ke saath kaise evolve hote hain.
Constraints and Lagrangian Mechanics — DOF = generalised coordinates ki count.
Equipartition Theorem — har DOF mein $\frac{1}{2} k_{B} T$ energy hoti hai (diatomics ke liye 5 use karta hai).
Rolling Motion — rolling-without-slipping constraint DOF reduce karta hai.