A metal rod if fixed rigidly at two ends so as to prevent its thermal expension. If L,`alpha` , Y respectively denote the length of the rod , coefficient of linear thermal expension and Young's modulus of its material, then for an increase in temperature of the rod by `Delta`T, the longitudinal stress developed in the rod is

Options

A. inversely proportional to `alpha`
B. inversely proportional to Y
C. directly proportinal to `Delta`T/Y
D. independent of L

Explanation

To solve the problem of determining the longitudinal stress developed in a metal rod that is fixed at both ends and subjected to an increase in temperature, we can follow these steps:

### Step 1: Understand the relationship between length, temperature change, and thermal expansion.
The change in length of a rod due to thermal expansion can be expressed as:
\[ L' = L(1 + \alpha \Delta T) \]
where:
- \( L' \) is the new length of the rod,
- \( L \) is the original length of the rod,
- \( \alpha \) is the coefficient of linear thermal expansion,
- \( \Delta T \) is the change in temperature.

### Step 2: Calculate the change in length (\( \Delta L \)).
The change in length (\( \Delta L \)) can be calculated as:
\[ \Delta L = L' - L \]
Substituting the expression for \( L' \):
\[ \Delta L = L(1 + \alpha \Delta T) - L \]
\[ \Delta L = L\alpha \Delta T \]

### Step 3: Relate the change in length to strain.
Strain (\( \epsilon \)) is defined as the change in length per unit original length:
\[ \epsilon = \frac{\Delta L}{L} \]
Substituting for \( \Delta L \):
\[ \epsilon = \frac{L\alpha \Delta T}{L} \]
Thus, we have:
\[ \epsilon = \alpha \Delta T \]
This is our Equation (1).

### Step 4: Use Hooke's Law to find stress.
According to Hooke's Law, the relationship between stress (\( \sigma \)), Young's modulus (\( Y \)), and strain (\( \epsilon \)) is given by:
\[ Y = \frac{\sigma}{\epsilon} \]
From this, we can express stress as:
\[ \sigma = Y \cdot \epsilon \]

### Step 5: Substitute the expression for strain into the stress equation.
Substituting Equation (1) into the stress equation:
\[ \sigma = Y \cdot (\alpha \Delta T) \]
Thus, we find that:
\[ \sigma = Y \alpha \Delta T \]

### Conclusion
The longitudinal stress developed in the rod due to an increase in temperature \( \Delta T \) is given by:
\[ \sigma = Y \alpha \Delta T \]