Question

A metal rod of length \( L \) and cross-sectional area \( A \) is heated through \( T \) °C. What is the force required to prevent the expansion of the rod lengthwise?
(Y = Young's modulus of material of the rod, \( \alpha \) = coefficient of linear expansion of the rod.)

• A. \( \frac{YA \alpha T}{(1 + \alpha T)} \)
• B. \( \frac{YA \alpha T}{(1 - \alpha T)} \)
• C. \( \frac{YA \alpha T}{(1 + \alpha T)} \)
• D. \( \frac{YA \alpha}{(1 - \alpha T)} \)

Hint:

When a material is heated, the force required to prevent its expansion is proportional to its Young's modulus, cross-sectional area, and the temperature change.

Answer 1

The Correct Option is C

Step 1: Understanding the force due to thermal expansion.
When a metal rod is heated, it expands. The force required to prevent this expansion is related to the Young's modulus and the change in length due to temperature. The formula for the force required to prevent the expansion is: \[ F = Y A \alpha T \left( \frac{1}{1 + \alpha T} \right) \] where \( Y \) is the Young's modulus, \( A \) is the cross-sectional area, \( \alpha \) is the coefficient of linear expansion, and \( T \) is the temperature change.
Step 2: Conclusion.
Thus, the correct answer is (C) \( \frac{YA \alpha T}{(1 + \alpha T)} \).