A meter scale calibarated for temperature T is used to measure the length of a steel rod. Length of steel rod measured by metal scale at temperature 2T is L. Length of the rod measured by metal scale at temperature T is `[alpha_(s)`:Coefficient of linear expansion of metal scale, `alpha_(R )` : Coefficient of linear expansion of steel rod]

Options

A. `(L)/(1+alpha_(s)T)`
B. `L(bot+alpha_(R)T)`
C. `(Lbot+alpha_(R)T)/((bot+alpha_(S)T))`
D. `(L(1+alpha_(S)T))/((1+alpha_(R)T))`

Explanation

To solve the problem, we need to understand the relationship between the lengths measured at different temperatures, taking into account the coefficients of linear expansion for both the steel rod and the metal scale.

### Step-by-Step Solution:

1. **Define the Lengths at Different Temperatures**:
- Let \( L_0 \) be the actual length of the steel rod at temperature \( T \).
- At temperature \( 2T \), the length of the steel rod, due to thermal expansion, becomes:
\[ L' = L_0 (1 + \alpha_R (2T - T)) = L_0 (1 + \alpha_R T) \]
- Here, \( \alpha_R \) is the coefficient of linear expansion of the steel rod.

2. **Measure the Length with the Metal Scale**:
- The length of the rod measured with the metal scale at temperature \( 2T \) is given as \( L \).
- The metal scale itself will also expand due to the increase in temperature. The change in length of the metal scale can be expressed as:
\[ \Delta L_{scale} = L_0 \alpha_S (2T - T) = L_0 \alpha_S T \]
- Therefore, the measured length \( L \) at temperature \( 2T \) is:
\[ L = L' - \Delta L_{scale} = L_0 (1 + \alpha_R T) - L_0 \alpha_S T \]
- Simplifying this gives:
\[ L = L_0 (1 + \alpha_R T - \alpha_S T) = L_0 (1 + (\alpha_R - \alpha_S) T) \]

3. **Relate Measured Length to Actual Length**:
- Rearranging the equation gives us the actual length \( L_0 \):
\[ L_0 = \frac{L}{1 + (\alpha_R - \alpha_S) T} \]

4. **Final Expression**:
- The expression for the actual length of the steel rod at temperature \( T \) in terms of the measured length \( L \) at temperature \( 2T \) is:
\[ L_0 = \frac{L}{1 + (\alpha_R - \alpha_S) T} \]