First slide

Thermal expansion

Question

Three rods of equal length l are joined to form an equilateral triangle PQR. O is the mid-point of PQ. Distance OR remains same for small change in temperature. Coefficient of linear expansion for PR and RQ is same, i.e. $α_{2}$ but that for PQ is $α_{1}$ . Then,

Difficult

A

$α_{2} = 3 α_{1}$
B

$α_{2} = 4 α_{1}$
C

$α_{1} = 3 α_{2}$
D

$α_{1} = 4 α_{2}$

Solution

$(O R)^{2} = (P R)^{2} - (P O)^{2} = l^{2} - {(\frac{l}{2})}^{2}$
$\Rightarrow$ $(O R)^{2} = {[l (1 + α_{2} t)]}^{2} - {[\frac{l}{2} (1 + α_{1} t)]}^{2}$
$l^{2} - \frac{l^{2}}{4} = l^{2} (1 + α_{2}^{2} t^{2} + 2 α_{2} t) - \frac{l^{2}}{4} (1 + α_{1}^{2} t^{2} + 2 α_{1} t)$
Neglecting $α_{2}^{2} t^{2}$ and $α_{1}^{2} t^{2}$ , we get
$0 = l^{2} (2 α_{2} t) - \frac{l^{2}}{4} (2 α_{1} t)$
$\Rightarrow$ $2 α_{2} = \frac{2 α_{1}}{4}$
$\Rightarrow$ $α_{1} = 4 α_{2}$

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