A metal rod of length `100cm`, made of silver at `0^(@)C` is heated to `100^(@)C` . It's length is increased by `0.19 cm` . Coefficient of cubical expansion of the silver rod is

Options

A. `5.7xx10^(-5)//^(@)C`
B. `0.63xx10^(-5)//^(@)C`
C. `1.9xx10^(-5)//^(@)C`
D. `16.1xx10^(-5)//^(@)C`

Explanation

To find the coefficient of cubical expansion of the silver rod, we can follow these steps:

### Step 1: Identify the given values
- Length of the rod at 0°C, \( L = 100 \, \text{cm} \)
- Increase in length, \( \Delta L = 0.19 \, \text{cm} \)
- Initial temperature, \( T_1 = 0°C \)
- Final temperature, \( T_2 = 100°C \)

### Step 2: Calculate the change in temperature
The change in temperature, \( \Delta T \), can be calculated as:
\[ \Delta T = T_2 - T_1 = 100°C - 0°C = 100°C \]

### Step 3: Use the formula for linear expansion
The formula for linear expansion is given by:
\[ \Delta L = \alpha \cdot L \cdot \Delta T \]
Where:
- \( \Delta L \) is the change in length,
- \( \alpha \) is the coefficient of linear expansion,
- \( L \) is the original length,
- \( \Delta T \) is the change in temperature.

### Step 4: Rearrange the formula to solve for \( \alpha \)
Rearranging the formula to find \( \alpha \):
\[ \alpha = \frac{\Delta L}{L \cdot \Delta T} \]

### Step 5: Substitute the known values into the equation
Substituting the values we have:
\[ \alpha = \frac{0.19 \, \text{cm}}{100 \, \text{cm} \cdot 100°C} = \frac{0.19}{10000} = 0.000019 \, \text{cm/°C} \]
Converting this to standard scientific notation:
\[ \alpha = 1.9 \times 10^{-5} \, \text{per °C} \]

### Step 6: Find the coefficient of cubical expansion \( \gamma \)
The relationship between the coefficient of linear expansion \( \alpha \) and the coefficient of cubical expansion \( \gamma \) is:
\[ \gamma = 3\alpha \]
Substituting the value of \( \alpha \):
\[ \gamma = 3 \times (1.9 \times 10^{-5}) = 5.7 \times 10^{-5} \, \text{per °C} \]

### Step 7: Conclusion
The coefficient of cubical expansion of the silver rod is:
\[ \gamma = 5.7 \times 10^{-5} \, \text{per °C} \]

### Final Answer:
The correct option is option 1: \( 5.7 \times 10^{-5} \, \text{per °C} \).

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