The correct option is
B α1α2<Y22Y1Given:
For wire
A,
Coefficient of thermal expansion
=α1 Young's modulus
=Y1 For wire
B,
Coefficient of thermal expansion
=α2 Young's modulus
=Y2 Initial temperature,
T=0 ∘C For case-1,
Temperature upto which wire
A is heated
=T1 Temperature upto which wire
B is cooled
=T2 For case-2,
Temperature upto which only wire
A is cooled
=T2 F.B.D of wire
A:
Here,
F1 is the force on wire
A due to its expansion on heating to temperature
T1 and
F2 is the force on wire
A due to contraction on cooling of wire
B to temperature
T2. Also
F′1 is the force on wire
A due to contraction on cooling of wire
A to temperature
T2.
So, in case 1, net tension is
F2−F1=Y2A2α2T2−Y1A1α1T1 .....(1)
[ from formula of linear expansion and Young's modulus ]
and in case 2, net tension is
F′1=Y1A1α1T2 .......(2)
[taking rightwards as positive]
Given, tension in both cases are equal.
From (1) and (2),
⇒Y2A2α2T2−Y1A1α1T1=Y1A1α1T2 ⇒Y2A2α2T2=Y1A1α1(T1+T2) ⇒(T1+T2)T2=Y2A2α2Y1A1α1 On applying componendo and dividendo, if
ab=cd then
a−2bb=c−2dd ⇒T1−T2T2=Y2A2α2−2Y1A1α1Y1A1α1 ⇒ΔTT2=Y2α2−2Y1α1Y1α1 [ given cross sectional area are equal ]
As
T1 is positive and
T2 is negative, so,
ΔT=T1−(−T2)= positive
Hence,
Y2α2−2Y1α1>0 ⇒α1α2<Y22Y1