A metal of length L whose coefficient of linear expansion α = 1 0 − 3 / ∘ C is heated such that it's temperature changes by 1000 K . Assuming its α is constant during the temperature change. [Tak

Question

A metal of length L whose coefficient of linear expansion α = 1 0 − 3 / ∘ C is heated such that it's temperature changes by 1000 K . Assuming its α is constant during the temperature change. [Take e ≈ 2.7]

TestHub · Accepted Answer

The coefficient of linear expansion is defined as $α = \frac{1}{L} \frac{d L}{d T}$ .

Given $α = 1 0^{- 3} /^{\circ} C$ and $Δ T = 1000 K$ .

Since $1 K = 1^{\circ} C$ , $Δ T = 100 0^{\circ} C$ .

Rearranging the formula, we get $\frac{d L}{L} = α d T$ .

Integrating both sides:

$\int_{L_{0}}^{L_{f}} \frac{d L}{L} = \int_{T_{0}}^{T_{f}} α d T$

$ln (L_{f}) - ln (L_{0}) = α (T_{f} - T_{0})$

$ln (\frac{L _{f}}{L _{0}}) = α Δ T$

$\frac{L _{f}}{L _{0}} = e^{α Δ T}$

Given $L_{0} = L$ , $α = 1 0^{- 3} /^{\circ} C$ , and $Δ T = 100 0^{\circ} C$ .

$\frac{L _{f}}{L} = e^{(1 0^{- 3} \times 1000)}$

$\frac{L _{f}}{L} = e^{1}$

Using the approximation $e \approx 2.7$ :

$\frac{L _{f}}{L} = 2.7$

$L_{f} = 2.7 L$

The final length of the rod is $2.7 L$ .

The increase in length is $Δ L = L_{f} - L = 2.7 L - L = 1.7 L$ .

Let's evaluate the options:

A. Final length of the rod is greater than $2 L$ .

$2.7 L > 2 L$ . This is TRUE.

B. Final length of the rod is greater than $2.5 L$ .

$2.7 L > 2.5 L$ . This is TRUE.

C. Final length of the rod is greater than $3 L$ .

$2.7 L > 3 L$ . This is FALSE.

D. Increase in the length due to heating is $L$ .

Increase in length is $1.7 L$ , which is not $L$ . This is FALSE.

The correct options are A and B.

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