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

Moderate

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

$\begin{array}{l} (OR)^{2} = (PR)^{2} - (PO)^{2} = l^{2} - {(\frac{l}{2})}^{2} \\ {[l (1 + α_{2} t)]}^{2} - {[\frac{1}{2} (1 + α_{1} t)]}^{2} \\ l^{2} - \frac{l^{2}}{4} = l^{2} (1 + a_{2}^{2} t^{2} + 2 α_{2} t) - \frac{l^{2}}{4} (1 + a_{1}^{2} t^{2} + 2 α_{1} t) \end{array}$
Neglecting $α_{2}^{2} t^{2} and α_{1}^{2} t^{2}$
$0 = l^{2} (2 α_{2} t) - \frac{l^{2}}{4} (2 α_{1} t) \Rightarrow 2 α_{2} = \frac{2 α_{1}}{4} \Rightarrow; α_{1} = 4 α_{2}$

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