A steel tape placed around the earth at the equator when the temperature is `10^(@)C`. What will be the clearance between the tape and the ground (assumed to be uniform) if the temperature of the tape rises to `40^(@)C`? Neglect expansion of the earth. Radius of earth at equator is `6400`km & ` alpha_(steel)= 1.2 xx 10^(-5)K^(-1)`

Options

A. `2.3` m
B. `2.1 ` m
C. `2.3`km
D. `230` m

Explanation

To solve the problem, we need to calculate the clearance between a steel tape placed around the Earth at the equator when the temperature rises from \(10^\circ C\) to \(40^\circ C\). We will use the formula for linear expansion to find the change in radius of the tape.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Initial temperature, \(T_1 = 10^\circ C\)
- Final temperature, \(T_2 = 40^\circ C\)
- Radius of the Earth at the equator, \(R_0 = 6400 \text{ km} = 6.4 \times 10^6 \text{ m}\)
- Coefficient of linear expansion for steel, \(\alpha = 1.2 \times 10^{-5} \text{ K}^{-1}\)

2. **Calculate the Change in Temperature:**
\[ \Delta T = T_2 - T_1 = 40^\circ C - 10^\circ C = 30 \text{ K} \]

3. **Use the Linear Expansion Formula:**
The change in length (or radius in this case) due to thermal expansion can be calculated using the formula:
\[ \Delta R = R_0 \cdot \alpha \cdot \Delta T \]

4. **Substitute the Values:**
\[ \Delta R = (6.4 \times 10^6 \text{ m}) \cdot (1.2 \times 10^{-5} \text{ K}^{-1}) \cdot (30 \text{ K}) \]

5. **Calculate \(\Delta R\):**
\[ \Delta R = 6.4 \times 10^6 \cdot 1.2 \times 10^{-5} \cdot 30 \]
\[ \Delta R = 6.4 \times 1.2 \times 30 \times 10^1 \]
\[ \Delta R = 2304 \text{ m} = 2.304 \text{ km} \]

6. **Conclusion:**
The clearance between the tape and the ground is approximately \(2.304 \text{ km}\).

### Final Answer:
The clearance between the tape and the ground is \(2.3 \text{ km}\).