First slide

Thermal expansion

Question

A copper rod of length 88 cm and an aluminium rod of unknown length have their increase in length independent of increase in temperature. The length of aluminium rod is (Here $α_{c u} = 17 \times 10^{- 5} p e r K a n d α_{A l} = 22 \times 10^{- 5} p e r K$ ) . [NEET (National) 2019]

Easy

A

113.9 cm
B

88 cm
C

68 cm
D

6.8 cm

Solution

Due to change in temperature, the thermal strain produced in a rod of length L is given by
   $\frac{Δ L}{L} = α Δ T \Rightarrow Δ L = L α Δ T$
where, L = original length of rod and $α$ = coefficient of linear expansion of solid rod.
As, the change in length ( $∆$ l) of the two given rods of copper and aluminium are independent of temperature change, i.e. $∆$ T is same for both copper and aluminium.
   $L_{C u} α_{C u} = L_{A l} α_{A l}$ …(i)
Here,    $α_{C u} = 17 \times 10^{- 5} K^{- 1}$
   $α_{A l} = 22 \times 10^{- 5} K^{- 1}$
and    $L_{C u} = 88 c m$
Substituting the given values in Eq. (i), we get
   $L_{Al} = \frac{α_{Cu} L_{Cu}}{α_{Al}} = \frac{17 \times 10^{- 5} \times 88}{22 \times 10^{- 5}} ≃ 68 cm$

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