A barometer tube reads 76 cm of mercury. If the tube is gradually inclined at an angle of `60` with vertical, keeping the open end immersed in the mercury reservoir, the length of the mercury column will be

Options

A. 152 cm
B. 76 cm
C. 38 cm
D. `38 sqrt3 cm`

Answer

To solve the problem of determining the length of the mercury column in an inclined barometer tube, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have a barometer tube that reads 76 cm of mercury when vertical. When the tube is inclined at an angle of 60 degrees with the vertical, we need to find the new length of the mercury column.

2. **Identify the Variables**:
- Let \( S_v \) = height of the mercury column in the vertical tube = 76 cm.
- Let \( S_i \) = height of the mercury column in the inclined tube.
- Let \( H \) = vertical height of the mercury column when inclined.
- Angle of inclination \( \theta = 60^\circ \).

3. **Use the Relationship Between Heights**:
- In the vertical tube, the pressure is given by the formula \( P = \rho g h \), where \( h \) is the vertical height of the mercury column.
- For the inclined tube, the vertical height \( H \) can be related to the inclined length \( S_i \) using trigonometry:
\[ H = S_i \cos(\theta) \]
- Thus, we can rearrange this to find \( S_i \):
\[ S_i = \frac{H}{\cos(\theta)} \]

4. **Set Up the Pressure Equality**:
- The pressure exerted by the mercury column must be equal in both cases:
\[ \rho g H = \rho g S_v \]
- Since \( \rho g \) cancels out, we have:
\[ H = S_v \]
- Therefore, \( H = 76 \, \text{cm} \).

5. **Calculate \( S_i \)**:
- Substitute \( H \) into the equation for \( S_i \):
\[ S_i = \frac{H}{\cos(60^\circ)} \]
- We know that \( \cos(60^\circ) = \frac{1}{2} \):
\[ S_i = \frac{76 \, \text{cm}}{\frac{1}{2}} = 76 \, \text{cm} \times 2 = 152 \, \text{cm} \]

6. **Conclusion**:
- The length of the mercury column when the tube is inclined at 60 degrees is \( S_i = 152 \, \text{cm} \).

### Final Answer:
The length of the mercury column in the inclined barometer tube is **152 cm**.