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 find the length of the steel rod measured by the meter scale at temperature T, given that the length of the rod measured at temperature 2T is L. We will also consider the coefficients of linear expansion for both the steel rod and the meter scale.

### Step-by-step Solution:

1. **Define Variables:**
- Let \( L_0 \) be the actual length of the steel rod at temperature \( T \).
- Let \( L' \) be the length of the steel rod measured at temperature \( 2T \) using the meter scale.
- We know that \( L' = L \) (the measured length at \( 2T \)).

2. **Length of Steel Rod at Temperature \( 2T \):**
- The change in length of the steel rod when the temperature changes from \( T \) to \( 2T \) can be expressed as:
\[ L' = L_0 + \Delta L_R \]
- Where \( \Delta L_R \) is the change in length of the steel rod, given by:
\[ \Delta L_R = L_0 \cdot \alpha_R \cdot (2T - T) = L_0 \cdot \alpha_R \cdot T \]
- Therefore, we can write:
\[ L' = L_0 (1 + \alpha_R T) \]

3. **Length of Meter Scale at Temperature \( 2T \):**
- The meter scale is calibrated for temperature \( T \) but is used at temperature \( 2T \). The change in length of the meter scale is:
\[ \Delta L_S = L \cdot \alpha_S \cdot (2T - T) = L \cdot \alpha_S \cdot T \]
- The actual length of the rod measured by the scale at \( 2T \) is:
\[ L' = L + \Delta L_S \]
- Thus, we can express this as:
\[ L' = L + L \cdot \alpha_S \cdot T = L(1 + \alpha_S T) \]

4. **Equating the Two Expressions:**
- Since both expressions for \( L' \) represent the same physical length of the rod, we can equate them:
\[ L_0 (1 + \alpha_R T) = L (1 + \alpha_S T) \]

5. **Solving for \( L_0 \):**
- Rearranging the equation to solve for \( L_0 \):
\[ L_0 = \frac{L (1 + \alpha_S T)}{(1 + \alpha_R T)} \]

### Final Answer:
The length of the steel rod measured by the meter scale at temperature \( T \) is:
\[ L_0 = \frac{L (1 + \alpha_S T)}{(1 + \alpha_R T)} \]