First slide

Thermal expansion

Question

A rod of length 20 cm is made of metal. It expands by 0.075cm when its temperature is raised from 0^oC to 100^oC. Another rod of a different metal B having the same length expands by 0.045 cm for the same change in temperature. A third rod of the same length is composed of two parts, one of metal A and the other of metal B. This rod expands by 0.060 cm for the same change in temperature. The portion made of metal A has the length

Moderate

A

20 cm
B

10 cm
C

15 cm
D

18 cm

Solution

$ΔL = L_{0} αΔθ$
$Rod A : 0 .075 = 20 \times α_{A} \times 100 \Rightarrow α_{A} = \frac{75}{2} \times 10^{- 6} /^{\circ} C$
$rod B : 0 .045 = 20 \times α_{B} \times 100 \Rightarrow α_{B} = \frac{45}{2} \times 10^{- 6} /^{\circ} C$
For composite rod: x cm of A and (20 - x) cm of B, we have
$\begin{array}{l} 0 .060 = {xα}_{A} \times 100 + (20 - x) α_{B} \times 100 \\ = x [\frac{75}{2} \times 10^{- 6} \times 100 + (20 - x) \times \frac{45}{2} \times 10^{- 6} \times 100] \end{array}$
On solving, we get x = 10 cm.

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