A metal rod of Young's modules Y and coefficient of thermal expansion `alpha` is held at its two ends such that its length remains constant. If its temperature is raised by `t^(@)C`, the linear stress developed in it is

Options

A. `Yalphat`
B. `(1)/(Yalphat)`
C. `(alphat)/(Y)`
D. `(Y)/(alphat)`

Answer

To solve the problem, we need to find the linear stress developed in a metal rod when its temperature is raised while its length is held constant. Here’s a step-by-step solution:

### Step 1: Understand the change in length due to temperature change
When the temperature of a rod is increased, it tends to expand. The change in length (ΔL) due to a temperature change (t) can be expressed as:
\[ \Delta L = L \cdot \alpha \cdot t \]
where:
- \(L\) is the original length of the rod,
- \(\alpha\) is the coefficient of thermal expansion,
- \(t\) is the change in temperature in degrees Celsius.

### Step 2: Recognize that the length remains constant
Since the rod is held at both ends, its length remains constant. This means that any tendency to expand (ΔL) must be countered by an internal stress that develops in the rod.

### Step 3: Define strain
Strain (ε) is defined as the change in length per unit length:
\[ \text{Strain} = \frac{\Delta L}{L} \]
Substituting the expression for ΔL, we get:
\[ \text{Strain} = \frac{L \cdot \alpha \cdot t}{L} = \alpha \cdot t \]

### Step 4: Relate stress to strain using Young's modulus
According to Hooke's Law, stress (σ) is related to strain (ε) by Young's modulus (Y):
\[ \sigma = Y \cdot \text{Strain} \]
Substituting the expression for strain, we have:
\[ \sigma = Y \cdot (\alpha \cdot t) \]

### Step 5: Final expression for stress
Thus, the linear stress developed in the rod when its temperature is raised by \(t\) degrees Celsius while its length remains constant is given by:
\[ \sigma = Y \cdot \alpha \cdot t \]

### Conclusion
The final answer is:
\[ \sigma = Y \cdot \alpha \cdot t \]