Coefficient of linear expansion of brass and steel rods are α1 and α2 respectively. Lengths of brass and steel rods are l1 and l2 respectively. If (l2 - l1) is maintained the same... Coefficient of linear expansion of brass and steel rods are α1 and α2 respectively. Lengths of brass and steel rods are l1 and l2 respectively. If (l2 - l1) is maintained the same at all temperatures, which one of the following relations holds good?
Understand the Problem
The question is asking for a relation between the coefficients of linear expansion and lengths of two rods made of different materials (brass and steel) while maintaining certain conditions. This involves concepts of thermal expansion in physics.
Answer
The relation that holds good is: $$ \alpha_1 L_1 = \alpha_2 L_2 $$
Answer for screen readers
The correct relation is:
$$ \alpha_1 L_1 = \alpha_2 L_2 $$
Steps to Solve
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Understanding Linear Expansion
The linear expansion of a material is given by the formula:
$$ \Delta L = \alpha L_0 \Delta T $$
where $\Delta L$ is the change in length, $\alpha$ is the coefficient of linear expansion, $L_0$ is the original length, and $\Delta T$ is the change in temperature. -
Set Up the Change in Length for Brass and Steel Rods
For the brass rod (length $L_1$ and coefficient $\alpha_1$), the change in length is:
$$ \Delta L_1 = \alpha_1 L_1 \Delta T $$
For the steel rod (length $L_2$ and coefficient $\alpha_2$), the change in length is:
$$ \Delta L_2 = \alpha_2 L_2 \Delta T $$ -
Relation Given in the Problem
The problem states that the difference in lengths $(L_2 - L_1)$ is maintained the same, which means:
$$ L_2 + \Delta L_2 = L_1 + \Delta L_1 $$
Rearranging gives us:
$$ \Delta L_2 - \Delta L_1 = L_1 - L_2 $$ -
Substituting the Change in Length Expressions
Substituting for $\Delta L_1$ and $\Delta L_2$ from the previous equations, we have:
$$ \alpha_2 L_2 \Delta T - \alpha_1 L_1 \Delta T = 0 $$
Factoring out $\Delta T$ gives:
$$ \Delta T (\alpha_2 L_2 - \alpha_1 L_1) = 0 $$ -
Implication of the Expression
Since $\Delta T$ cannot be zero (as it represents temperature change), we obtain the relationship:
$$ \alpha_2 L_2 = \alpha_1 L_1 $$
This means that when the difference in lengths remains constant, the coefficients and original lengths of the materials are related through this equation.
The correct relation is:
$$ \alpha_1 L_1 = \alpha_2 L_2 $$
More Information
This relationship is rooted in the principles of thermal expansion in materials. When two rods of different materials are subjected to temperature changes, their expansion will depend on their respective coefficients and original lengths. This relation is often used in engineering to ensure stability in structures involving different materials.
Tips
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Forgetting to factor out $\Delta T$: Not recognizing that $\Delta T$ is non-zero can lead to incorrect conclusions. Always ensure you account for temperature changes.
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Misapplying formulas: Misunderstanding the formula for linear expansion can result in using the wrong coefficients or lengths. Always verify your formulas.
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