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 ?

Options

A. `(YAalphaT)/(1-alphaT)`
B. `(YAalphaT)/(1+alphaT)`
C. `(1-alphaT)/(YAalphaT)`
D. `((1+alphaT))/(YAalphaT)`

Answer

To find the force required to prevent the expansion of a metal rod of length \( L \) and cross-sectional area \( A \) when heated through \( T \) degrees Celsius, we can follow these steps:

### Step 1: Understand the Thermal Expansion
When the rod is heated, it expands. The increase in length \( \Delta L \) due to thermal expansion can be expressed as:
\[ \Delta L = \alpha L T \]
where \( \alpha \) is the coefficient of linear expansion of the material.

### Step 2: Determine the New Length
The new length \( L' \) of the rod after heating can be expressed as:
\[ L' = L + \Delta L = L + \alpha L T = L(1 + \alpha T) \]

### Step 3: Apply Compressive Force
To prevent the rod from expanding, we need to apply a compressive force \( F \). This force will create a compressive stress in the rod, which can be expressed as:
\[ \text{Stress} = \frac{F}{A} \]

### Step 4: Relate Stress to Strain
According to Hooke's Law, the stress is also related to strain by Young's modulus \( Y \):
\[ \text{Stress} = Y \times \text{Strain} \]
The strain \( \text{Strain} \) can be defined as:
\[ \text{Strain} = \frac{\Delta L}{L'} \]
Substituting \( \Delta L = \alpha L T \) and \( L' = L(1 + \alpha T) \):
\[ \text{Strain} = \frac{\alpha L T}{L(1 + \alpha T)} = \frac{\alpha T}{1 + \alpha T} \]

### Step 5: Set Up the Equation
Now, we can equate the expressions for stress:
\[ \frac{F}{A} = Y \times \frac{\alpha T}{1 + \alpha T} \]

### Step 6: Solve for the Force \( F \)
Rearranging the equation to solve for the force \( F \):
\[ F = A Y \times \frac{\alpha T}{1 + \alpha T} \]

### Final Expression
Thus, the force required to prevent the expansion of the rod lengthwise is:
\[ F = \frac{A Y \alpha T}{1 + \alpha T} \]