A student records the initial length $l$, change in temperature $\Delta T$ and change in length $\Delta l$ of a rod as follows :

S.No. $l(m)$ $\Delta T{(^o}C)$ $\Delta l(m)$
$(1)$ $2$ $10$ $4\times 10^{-4}$
$(2)$ $1$ $10$ $4\times 10^{-4}$
$(3)$ $2$ $20$ $2\times 10^{-4}$
$(4)$ $3$ $10$ $6\times 10^{-4}$

If the first observation is correct, what can you say about observations  $2,\,3$ and $4$.

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From first observation,

$\alpha=\frac{\Delta l}{l \Delta \mathrm{T}}$

$=\frac{4 \times 10^{-4}}{2 \times 10}$

$=2 \times 10^{-5}{ }^{\circ} \mathrm{C}^{-1}$

From second observation,

$\Delta l= \alpha l \Delta \mathrm{T}$

$\therefore \Delta l=2 \times 10^{-5} \times 1 \times 10$

$=2 \times 10^{-4} \mathrm{~m} \neq 4 \times 10^{-4} \mathrm{~m}$

$\therefore$ Thus, the observation is incorrect.

From third observation,

$\Delta l=\alpha l \Delta \mathrm{T}$

$=2 \times 10^{-5} \times 2 \times 20$

$=8 \times 10^{-4} \mathrm{~m} \neq 2 \times 10^{-4} \mathrm{~m}$

$\therefore$ Thus, the observation is incorrect.

From fourth observation,

$\Delta l=\alpha l \Delta \mathrm{T}$

$=2 \times 10^{-5} \times 3 \times 10$

$=6 \times 10^{-4} \mathrm{~m}=6 \times 10^{-4} \mathrm{~m}$

Thus, the observation is true.

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