1.5.1 · D5 · HinglishRotational Mechanics
Question bank — Rigid body — definition, degrees of freedom
1.5.1 · D5· Physics › Rotational Mechanics › Rigid body — definition, degrees of freedom
Recurring master formula jis par tum rely karoge:
True or false — justify karo
True or false: Ek rigid body jisme particles hain, uske degrees of freedom hain.
False. Free particles dete, lekin rigidity saari inter-particle distances lock kar deti hai; 3 non-collinear points pin hone ke baad (), har doosra particle force ho jaata hai, isliye DOF par saturate ho jaati hai.
True or false: Body ko "zyada rigid" banana uske degrees of freedom ka number badalta hai.
False. Ideal model mein rigidity all-or-nothing hoti hai — ek baar distances freeze ho jayein, toh DOF hi rehti hai chahe material kitna bhi stiff ho. Stiffness ek material property hai, DOF ek counting result hai.
True or false: Ek rigid body jo 2D plane mein confined hai, uske DOF hain.
False. Ek plane mein centre ko sirf chahiye aur sirf ek rotation axis hoti hai (plane ke bahar), jo DOF deti hai, nahi.
True or false: Ek rigid body ke liye independent constraints ki sankhya hoti hai.
False. Har pair ka ek fixed-distance equation hota hai, lekin ke liye wo equations redundant ho jaate hain; independent constraints ki sankhya hoti hai.
True or false: Ek diatomic molecule jisme do point atoms hain, uske DOF hain.
False. Point masses ko apni connecting line ke baare mein spin karna kuch move nahi karta, isliye woh rotational DOF physically absent hai: DOF.
True or false: Ek rigid body ke ek point ko fix karne se exactly DOF remove hoti hain.
True. Ek point fix karna teeno translations khatam kar deta hai lekin us point ke baare mein teeno rotations intact rehti hain, isliye bachti hain.
True or false: Fixed axle par ek wheel aur ek swinging door ke DOF ki sankhya same hoti hai.
True. Dono ka ek fixed axis hai, sirf ek spin/swing angle bachta hai, isliye dono ke exactly DOF hai.
True or false: Agar tum ek rigid body ke kisi bhi particles ki positions jaante ho, toh tum unka sab ka position jaante ho.
True — provided woh 3 non-collinear hain. Unki har doosre particle se fixed distances phir har baaki particle ko uniquely pin kar deti hain; collinear points fail ho jaate hain kyunki woh us line ke baare mein rotation fix nahi karte.
Error dhundo
Error dhundo: "Ek rigid body ke DOF hote hain, toh ek -particle body ke DOF hote hain."
Jo formula quote kiya gaya hai woh constraints ke liye hai, DOF ke liye nahi. DOF hamesha hoti hai; independent constraints count karta hai (yahan ), aur .
Error dhundo: "Ek rigid body ko fix karne ke liye mujhe points fix karne chahiye, aur koi bhi points chalenge."
Woh points non-collinear hone chahiye. Ek straight line par teen points body ko us line ke baare mein rotate karne ke liye free chodh dete hain, isliye orientation fully fix nahi hoti.
Error dhundo: "Ek rigid body ke hamesha DOF hote hain, period."
Ye sirf tab sach hai jab 3D mein free ho. Constraints isko reduce karte hain: fixed axis , fixed point , ek plane mein confined . Hamesha pehle count karo kya constrained hai.
Error dhundo: "3-point derivation mein Point 2, Point 1 ki tarah coordinates contribute karta hai."
Point 2, Point 1 se ek fixed distance par constrained hai, isliye woh ek sphere par rehta hai aur sirf independent coordinates contribute karta hai ().
Error dhundo: "Ek rod jo load ke neeche thoda bend karta hai, woh rigid body nahi hai, isliye hum uske liye rigid-body DOF use nahi kar sakte."
Rigid body ek idealization hai; chhoti deformations drop kar di jaati hain (80/20 move). Ordinary forces ke liye hum rod ko phir bhi rigid treat karte hain aur DOF use karte hain kyunki bending negligible hai.
Error dhundo: " translational DOF describe karte hain ki body kaise spin karti hai."
Translation describe karta hai kahan body baithti hai ( ke saath bina ghume slide karna). Spinning rotational DOF (roll, pitch, yaw) se capture hoti hai.
Error dhundo: "Kyunki ek diatomic ek rotation khota hai, ek triatomic bent molecule ke bhi DOF hote hain."
Ek non-linear (bent) triatomic collinear nahi hai, isliye har axis ke baare mein rotation atoms ko move karta hai; uske saare rotations rehte hain, jo DOF deta hai. Sirf point masses ki linear arrangements ek rotation khoti hain.
Why questions
Hum directly fixed-distance equations count karne ke bajaye constraints subtract kyun karte hain?
Kyunki woh pair-distance equations sab independent nahi hote; blindly count karna over-count karta hai. se independent constraints () subtract karna correctly yield karta hai.
DOF count non-collinear points ke baad grow karna band kyun ho jaata hai?
aisi points position aur orientation dono fix kar lene ke baad, har agle particle ka location unse fixed distances ke through determine ho jaata hai, new DOF add karta hai.
Bond axis ke baare mein rotation ek diatomic molecule ke liye DOF count kyun nahi hota?
Atoms ko us axis par pade point masses treat kiya jaata hai, isliye line ke baare mein rotate karna har particle ko exactly wahi chhod deta hai jahan tha — koi configuration change nahi matlab koi independent coordinate nahi.
Ek rigid body ki DOF kyun split hoti hai aur kabhi, maano, nahi?
3D space mein exactly independent translation directions aur exactly independent rotation axes hote hain; koi chautha tarika slide karne ya ghoomne ka exist nahi karta, isliye split forced hai.
Ek body ko plane mein confine karne par sirf rotational DOF kyun bachti hai ki jagah?
Ek rotation body ko plane mein rakhna chahiye, aur sirf woh axis jo plane ke perpendicular hai wo karta hai; kisi in-plane axis ke baare mein rotate karna body ko plane se bahar tip kar deta.
Why must the master formula use independent coordinates, not just any coordinates?
Dependent coordinates constraints ke through doosron se compute kiye ja sakte hain, isliye woh koi naya information nahi rakhte; sirf independent numbers genuinely configuration pin down karte hain.
Edge cases
Edge case: Ek rigid body ke kitne DOF hote hain agar uske do points fix ho jayein?
DOF. Do points fix karna ek axis fix karta hai (unke through line), aur sirf baaki freedom us axis ke baare mein rotation hai — spin angle .
Edge case: Ek rigid body ka DOF kya hoga jab teen non-collinear points fix ho jayein?
DOF. Body completely frozen hai — position aur orientation dono fully determined hain, isliye koi independent coordinate nahi bachta.
Edge case: Kya ek single free point particle ke rotational DOF hote hain?
Nahi — ek point ki koi orientation nahi hoti, isliye uske sirf translational DOF hain aur rotational DOF.
Edge case: Kis ke liye pair-distances sab independent hote hain?
Exactly , jahan . ke liye, , isliye kuch pair-constraints redundant ho jaate hain.
Edge case: Ek rigid body table par frictionlessly slide karti hai lekin lift off bhi kar sakti hai — kitne DOF?
. Surface chodh sakna matlab effectively 3D mein free hai; table tab hi constraint banta hai jab woh body ko contact mein rehne force kare (jo ise par drop kar deta).
Edge case: Ek rigid rod (ek line par bahut saare particles) ke free in 3D hone par DOF ka kya hoga?
Uske DOF hote hain, nahi — linear molecule ki tarah, apni lambi axis ke baare mein rotation kisi (point-like) particle ko move nahi karta, isliye ek rotational DOF vanish ho jaata hai. Real thickness wala rod poore recover kar leta hai.
Recall Pure bank ka ek-line summary
Free particles dete hain; rigidity independent constraints remove karta hai, bachta hai 3D mein free body ke liye — phir har constraint (axis, point, plane, collinearity) us ko neeche chip karta hai, aur linear point-mass bodies quietly ek rotation khoti hain.