Question 717: Thermodynamics & KTG

Question

Two rods of different materials have lengths ' $l$ ' and ' $l_2$ ' whose coefficient of linear expansions are ' $\alpha_1$ ' and ' $\alpha_2$ ' respectively. If the difference between the two lengths is independent of temperature then

Accepted Answer

Let the original lengths of the rods be $l_1$ and $l_2$.
Let the coefficients of linear expansion be $\alpha_1$ and $\alpha_2$.
Step 1 (Write the expansion formula):
When temperature increases by $\Delta T$, the new lengths are:
First rod: $l_1' = l_1 (1 + \alpha_1 \Delta T)$
Second rod: $l_2' = l_2 (1 + \alpha_2 \Delta T)$
Step 2 (Find the difference in lengths):
Difference after expansion:
$ 	ext{Difference} = l_1'(T) - l_2'(T) = l_1 (1 + \alpha_1 \Delta T) - l_2 (1 + \alpha_2 \Delta T) $
Simplify:
$ = (l_1 - l_2) + (l_1 \alpha_1 - l_2 \alpha_2)\Delta T $
Step 3 (Condition given in the question):
The difference should be
independent of temperature
.
So coefficient of $\Delta T$ must be zero:
$ l_1 \alpha_1 - l_2 \alpha_2 = 0 $
$ l_1 \alpha_1 = l_2 \alpha_2 $
$ \frac{l_1}{l_2} = \frac{\alpha_2}{\alpha_1} $
Step 4 (Choose the correct option):
This matches
Option B
.
Final Answer:
$ \boxed{\frac{l_1}{l_2} = \frac{\alpha_2}{\alpha_1}} $
So, the correct answer is
Option B
.

Answer

$\alpha_1^2 l_1=\alpha_2^2 l_2$

Answer

$\frac{l_1}{l_2}=\frac{\alpha_2}{\alpha_1}$

Answer

$\frac{l_1}{l_2}=\frac{\alpha_1}{\alpha_2}$

Answer

$l_1^2 \alpha_2=l_2^2 \alpha_1$

Thermodynamics & KTG

Two rods of different materials have lengths ' l ' and ' l_2 ' whose coefficient of linear expansions are ' \alpha_1 ' and ' \alpha_2 ' respectively. If the difference between the two lengths is independent of temperature then

Choose the correct answer: