Question

Two rods of length ${\mathrm{l}}_1$ and ${\mathrm{l}}_2$ are made of materials whose coefficients of linear expansion are ${\mathrm{\alpha }}_1$ and ${\mathrm{\alpha }}_2$. If at a certain change in temperature, the change in length of the rods are the same, what is the ratio of the original lengths of the rods?

• A. ${\mathrm{\alpha }}_1:{\mathrm{\alpha }}_2$
• B. ${\mathrm{\alpha }}_2:{\mathrm{\alpha }}_1$
• C. 1:1
• D. Cannot be determined

Answer 1

The correct option is B

${\mathrm{\alpha }}_2:{\mathrm{\alpha }}_1$

Thermal expansion:

Thermal expansion is the phenomenon of an object expanding due to a change in temperature.

The change in length of the rod due to the change in temperature can be expressed as follows:

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

Here,

1. $∆\mathrm{l}$ is the change in length.
2. $\alpha$ is the coefficient of thermal expansion.
3. $∆\mathrm{T}$ is the change in temperature.

Since the change in length and change in temperature for the rods are the same, the equation can be written as follows:

$$\begin{gathered} {\mathrm{l}}_1{\mathrm{\alpha }}_1∆\mathrm{T}={\mathrm{l}}_2{\mathrm{\alpha }}_2∆\mathrm{T} \\ {\mathrm{l}}_1{\mathrm{\alpha }}_1={\mathrm{l}}_2{\mathrm{\alpha }}_2 \\ \frac{{\mathrm{l}}_1}{{\mathrm{l}}_2}=\frac{{\mathrm{\alpha }}_2}{{\mathrm{\alpha }}_1} \end{gathered}$$

Therefore, the ratio of the original lengths of the rods is ${\mathrm{\alpha }}_2:{\mathrm{\alpha }}_1$.

Hence, option (B) is correct.