Thermal expansion — linear, area, volumetric: Question bank

1.7.7 · D5 · Physics › Thermodynamics › Thermal expansion — linear, area, volumetric

True or false — justify karo

Recall

T/F: Ek metal plate mein cut kiya hua hole, plate garam hone par chota ho jaata hai. ::: False. Plate ka har point same fraction se radially baahir jaata hai, isliye hole ki rim bhi baahir jaati hai — hole usi $β$ se bada hota hai jitna solid metal hota.

Recall

T/F: Agar temperature rise $Δ T$ double kar do, toh stretch $Δ L$ bhi roughly double ho jaata hai. ::: True (first order tak). Kyunki $Δ L = L_{0} α Δ T$ mein $Δ T$ linear hai, $Δ T$ double karne se $Δ L$ double hoga — jab tak $α Δ T ≪ 1$ ho, taaki exact $e^{α Δ T}$ abhi bhi $1 + α Δ T$ ke paas ho.

Recall

T/F: $30^{\circ} C$ ka temperature difference $30 K$ ke difference ke barabar hai. ::: True. Dono scales mein sirf $273.15$ ka offset ka fark hai, aur kisi bhi difference mein offsets cancel ho jaate hain — isliye $Δ T (K) = Δ T (^{\circ} C)$ exactly.

Recall

T/F: Same metal ka ek solid block, ek flat sheet, aur ek thin rod — teeno ko same amount garam karo — kisi ek edge ke along same fractional amount se badhte hain. ::: True. Ek single direction mein har ek $(1 + α Δ T)$ factor se bada hota hai, shape se koi fark nahi padta — bas fark yeh hai ki block teen directions mein woh factor stack karta hai (volume), sheet do mein (area), rod ek mein (length).

Recall

T/F: $β = 2 α$ sirf ek approximation hai; exact area factor $(1 + α Δ T)^{2}$ hai. ::: True. Exactly, $A = A_{0} (1 + α Δ T)^{2} = A_{0} (1 + 2 α Δ T + α^{2} Δ T^{2})$ . Hum $α^{2} Δ T^{2} \sim 1 0^{- 10}$ drop kar dete hain, aur $β = 2 α$ milta hai.

Recall

T/F: Paani garam hone par hamesha expand karta hai. ::: False. $0$ se $4^{\circ} C$ ke beech paani garam hone par contract karta hai ( $γ < 0$ ) — yeh famous anomaly hai, dekho Anomalous expansion of water. $4^{\circ} C$ ke upar yeh normally behave karta hai.

Recall

T/F: Ek gas ka ek single material constant $γ$ hota hai, bilkul solid ki tarah. ::: False. Ideal gas mein constant pressure par fractional volume change per degree $1/ T$ hota hai, jo current absolute temperature par depend karta hai — yeh koi fixed material property nahi hai. Dekho Gas laws aur Kinetic theory of gases.

Recall

T/F: Different metals ki do rods, fixed walls ke beech end-to-end clamp ki hui hain — garam hone par dono mein same stress develop hoga. ::: False. Har ek ka apna $α$ hai, isliye har ek different amount expand karna chahta hai; har ek mein jo thermal stress develop hota hai woh uske $α$ aur stiffness (Young's modulus) dono par depend karta hai — dekho Stress and strain.

Error dhundo

Recall

"Bracket hai $L = L_{0} e^{α Δ T}$ , lekin kyunki $e^{x}$ curved hai aur $α Δ T$ tiny hai, accuracy ke liye $L = L_{0} (1 + α Δ T + \frac{1}{2} α^{2} Δ T^{2})$ use kar sakte hain." ::: $\frac{1}{2} α^{2} Δ T^{2}$ term $1 0^{- 10}$ order ka hai — bilkul negligible aur kisi bhi experimental precision se pare. Ise rakhna false precision hai; $L_{0} (1 + α Δ T)$ hi honest working form hai.

Recall

"Area ke liye $β = 2 α$ use karte hain, toh plate ke hole ke liye uska aadha use karna chahiye, kyunki hole empty hai." ::: Galat. Emptiness ka apna koi expansion nahi hota; hole apne aas-paas ke metal ko track karta hai. Iske area ke liye poora $β = 2 α$ use hota hai, bilkul waisa hi jaise hole usi metal se bhara ho.

Recall

"Rod expand karti hai, isliye iska density same rehta hai — same atoms, same mass, same volume." ::: Mass same rehta hai, lekin volume bada hota hai, isliye density girti hai. Density $= m / V$ aur garam hone par $V$ badhta hai, toh hot object thoda kam dense hota hai cold object se.

Recall

"Pendulum clock ki brass rod summer mein lambi ho jaati hai, isliye woh faster swing karti hai aur clock time gain karta hai." ::: Ulta hai. Period $T = 2 π L / g$ $L$ ke saath badhta hai, isliye har swing zyada time leti hai aur clock slow chalta hai. Dekho Pendulum and simple harmonic motion.

Recall

"Kelvin mein convert karte hue, $Δ T = 40 + 273 - (10 + 273) = 323 - 283 = 40 K$ , isliye main $Δ T$ nikalte waqt hamesha 273 add karta hoon." ::: Arithmetic se sahi number milta hai, lekin har temperature mein 273 add karna faaltu kaam hai — offsets cancel ho jaate hain. Seedha $40 - 10 = 30$ lo. (273 add karna sirf absolute temperatures ke liye matter karta hai, jaise gas laws mein.)

Recall

"Thermal expansion isliye hota hai kyunki garam karne se energy add hoti hai jo atoms ke beech extra material create karti hai." ::: Koi naya material nahi banta. Garam karne se atoms zyada vibrate karte hain, aur kyunki interatomic potential well asymmetric hai (squeeze side par steeper), average spacing baahir drift karti hai — same atoms, zyada room. Dekho Interatomic potential energy curve.

Why questions

Recall

Kyun expansion ki deep cause potential well ki asymmetry hai, na ki sirf "atoms zyada move karte hain"? ::: Agar well symmetric hoti, toh zyada vibration atom ko andar-baahir equally far swing karaati, aur average position unchanged rehti — koi expansion nahi. Sirf isliye kyunki well compression side par steeper hai, time-averaged position baahir shift hoti hai.

Recall

Kyun volume length se teen guna fast expand karta hai ( $γ = 3 α$ ) — unhe add karne ki bajaye? ::: Teeno edges har ek $(1 + α Δ T)$ factor se badhte hain, aur volume unka product hai: $(1 + α Δ T)^{3} \approx 1 + 3 α Δ T$ . "3" teen dimensions ka exponent hai, koi sum nahi.

Recall

Kyun hum raw $Δ L$ ki bajaye fractional change $Δ L / L_{0}$ use karte hain? ::: Fraction intrinsic material property hai — same rehta hai chahe rod millimetres mein measure ho ya kilometres mein. Raw $Δ L$ object ke size ke saath scale karta hai isliye woh clean constant nahi hai.

Recall

Kyun bridges aur railway tracks mein expansion joints zaroori hain? ::: Ek lamba span din-raat ya seasonal temperature swing mein real, measurable $Δ L = L_{0} α Δ T$ accumulate karta hai; bina gap ke ends apne supports mein push karte aur damaging thermal stress build hota — dekho Stress and strain.

Recall

Kyun ek tight metal lid ko hot water ke neeche rakh kar dhila kiya ja sakta hai? ::: Metal lid glass se faster garm hoti hai aur zyada expand karti hai (higher $α$ ), isliye iska inner rim baahir badhta hai, grip dhili ho jaati hai. Lid ka hole bada hota hai, chota nahi.

Recall

Kyun gases same temperature rise par solids se bahut zyada expand karti hain? ::: Solid mein atoms ek deep well mein bound hote hain aur sirf thoda edge apart hote hain. Gas mein molecules essentially free hote hain; $T$ badhane se unki kinetic energy directly badhti hai, aur fixed pressure par volume absolute temperature ke saath scale karta hai — dekho Kinetic theory of gases.

Edge cases

Recall

$Δ T = 0$ hone par $Δ L$ kya hoga? ::: Exactly zero. $Δ L = L_{0} α \cdot 0 = 0$ — koi temperature change nahi, koi expansion nahi. Kisi bhi expansion formula ka useful sanity check.

Recall

Ek ring garam ki jaati hai. Kya metal ki width (annulus thickness) hole ko badhne dene ke liye shrink karti hai? ::: Nahi. Metal khud har direction mein expand karta hai, isliye annulus bhi mota hota hai. Hole aur metal dono milke same fractional factor se bade hote hain.

Recall

Kya $γ$ kabhi negative ho sakta hai, aur physically iska kya matlab hai? ::: Haan — $0$ se $4^{\circ} C$ tak paani ka $γ < 0$ hota hai, matlab garam hone par woh shrink karta hai (aur thanda hone par expand). Isliye talabon mein ice upar banti hai. Dekho Anomalous expansion of water.

Recall

Ek thin ring ko kaat ke khola jaata hai (ab closed loop nahi) aur garam kiya jaata hai. Kya cut ends ke beech ka gap bada hoga ya chota? ::: Bada hoga. Ring ka har part, cut faces bhi, general expansion ke saath baahir jaata hai, isliye ends alag hote hain — ring apne aap ka ek stretched-out version ban jaati hai.

Recall

Agar $α Δ T$ chota nahi hota (maan lo bahut bada temperature swing), toh kaun sa formula use karna padega? ::: Exact wala, $L = L_{0} e^{α Δ T}$ , kyunki linear approximation $L_{0} (1 + α Δ T)$ sirf tab kaam karti hai jab $α Δ T ≪ 1$ ; bade exponents ke liye neglected higher-order terms negligible nahi rehte.

Recall

Do identical rods — ek uniformly garam ki gayi, ek sirf ek end se — same final average temperature. Same total expansion? ::: First order tak, haan: total $Δ L$ rod ke along average temperature rise par depend karta hai, isliye equal average rise se equal net expansion milti hai (second-order aur stress effects ignore karte hue). Halaanki, non-uniform rod internal thermal stress carry karti hai.

Connections

Interatomic potential energy curve — woh asymmetry jo expansion ko exist karaati hai.
Kinetic theory of gases — kyun gases solids se bahut zyada expand karti hain.
Stress and strain — kya hota hai jab expansion rokaa jaaye.
Anomalous expansion of water — negative- $γ$ exception.
Pendulum and simple harmonic motion — slow-clock trap.
Gas laws — jahan $Δ T$ absolute hona chahiye.