Question

A rectangular block is heated from $0^{\circ }\text{C}$ to ${100}^{\circ }\text{C}$. The percentage increase in its length is $0.10\%$. What will be the percentage increase in its volume?

• A. $0.03\%$
• B. $0.10\%$
• C. $0.30\%$
• D. $0.5\%$

Answer 1

The correct option is C $0.30\%$

Given,
Percentage increase in length of rectangular block = $0.1\%$
$\therefore$ Fractional change in length$\frac{\Delta l}{l}=0.001$
Change in temperature $\left(\Delta T\right)={100}^{\circ }C$
For linear expansion, we know that,
$\frac{\Delta l}{l}=\alpha \Delta T$
From the data given in the question, we can write that,
$0.001=\alpha \times 100$
$\Rightarrow \alpha ={10}^{-5}{/}^{\circ }\text{C}$
Further, for a solid material, we know that
$\gamma =3\alpha$
$\Rightarrow \gamma =3\times {10}^{-5}{/}^{\circ }\text{C}$
To find percentage change in volume for a small change in temperature $\Delta T$:
We know from volume expansion that,
$\frac{\Delta V}{V}=\gamma \times \left(\Delta T\right)$
$\therefore \frac{\Delta V}{V}\times 100=\gamma \times \left(\Delta T\right)\times 100=(3\times {10}^{-5})\times 100\times 100=0.30\%$
Hence, option (c) is the correct answer.