A glass rod when measured with a zinc scale, both being at `30^(@)C`, appears to be of length `100`cm . If the scale shows correct reading at `0^(@)C`, then the true length of glass rod at `30^(@)C` and `0^(@)C` are :-
(`alpha_("glass") = 8 xx 10^(-6)"^(@)C ^(-1), alpha_("zinc") = 26 xx 10^(-6) K^(-1)`)

Options

A. `100.054` cm , `100.054`cm
B. `100.078 "cm" , 100.078"cm"`
C. `100.078 cm , 100.054`cm
D. `100.054` cm , `100.078`cm

Explanation

To solve the problem, we need to determine the true lengths of the glass rod at both 30°C and 0°C. We will use the coefficients of linear expansion for glass and zinc to account for the changes in length due to temperature variations.

### Step 1: Determine the apparent length of the glass rod at 30°C

The apparent length of the glass rod measured with the zinc scale at 30°C is given as:
\[ L_{apparent} = 100 \, \text{cm} \]

### Step 2: Calculate the true length of the glass rod at 30°C

The formula for linear expansion is:
\[ L = L_0 (1 + \alpha \Delta T) \]
Where:
- \( L \) = length at the new temperature
- \( L_0 \) = original length (apparent length in this case)
- \( \alpha \) = coefficient of linear expansion
- \( \Delta T \) = change in temperature

For the glass rod measured with the zinc scale, we need to consider the expansion of the zinc scale because it shows the apparent length. The coefficient of linear expansion for zinc is given as:
\[ \alpha_{zinc} = 26 \times 10^{-6} \, \text{°C}^{-1} \]

The change in temperature (\( \Delta T \)) from 0°C to 30°C is:
\[ \Delta T = 30 - 0 = 30 \, \text{°C} \]

Now, substituting the values into the linear expansion formula:
\[ L_{true} = L_{apparent} \cdot \left(1 + \alpha_{zinc} \cdot \Delta T\right) \]
\[ L_{true} = 100 \cdot \left(1 + 26 \times 10^{-6} \cdot 30\right) \]
\[ L_{true} = 100 \cdot \left(1 + 0.00078\right) \]
\[ L_{true} = 100 \cdot 1.00078 \]
\[ L_{true} = 100.078 \, \text{cm} \]

### Step 3: Calculate the true length of the glass rod at 0°C

Now we will find the true length of the glass rod at 0°C. We will use the coefficient of linear expansion for glass:
\[ \alpha_{glass} = 8 \times 10^{-6} \, \text{°C}^{-1} \]

Using the formula for linear expansion again, we can express the true length at 0°C as:
\[ L_{0°C} = \frac{L_{true}}{1 + \alpha_{glass} \cdot \Delta T} \]
Where \( \Delta T \) is now from 30°C to 0°C:
\[ \Delta T = 30 - 0 = 30 \, \text{°C} \]

Substituting the values:
\[ L_{0°C} = \frac{100.078}{1 + 8 \times 10^{-6} \cdot 30} \]
\[ L_{0°C} = \frac{100.078}{1 + 0.00024} \]
\[ L_{0°C} = \frac{100.078}{1.00024} \]
\[ L_{0°C} \approx 100.054 \, \text{cm} \]

### Final Results

- The true length of the glass rod at 30°C is approximately **100.078 cm**.
- The true length of the glass rod at 0°C is approximately **100.054 cm**.