First slide

Thermal expansion

Question

A steel scale measures the length of a copper wire as 80.0 cm when both are at 20°C (the calibration temperature for scale). What would be the scale read for the length of the wire when both are at 40°C? (Given $α_{steel}$ = $11 \times 10^{- 6} per ° C$ and $α_{copper} = 17 \times 10^{- 6} {per}^{0} C$ )

Easy

A

80.0096 cm
B

80.0272 cm
C

1 cm
D

25.5 cm

Solution

With temperature rise (same 25°C for both), steel scale and copper wire both expand. Hence length of copper wire w.r.t. steel scale or apparent length of copper wire after rise in temperature
$L_{app} = L_{0} + {L_{Cu}}^{'} - {L_{steel}}^{'}$
= $L_{0} +$ [ $L_{0} (1 + α_{Cu} ∆ θ) - L_{0} (1 + α_{s} ∆ θ)$ ]
$\Rightarrow L_{app} = L_{0} + L_{0} (α_{Cu} - α_{s}) ∆ θ$
=80+ 80(17 $\times 10^{- 6} - 11 \times 10^{- 6}) \times 20$ )
= 80.0096 cm

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