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. `38sqrt(3) cm`

Answer

To solve the problem of finding the length of the mercury column in a barometer tube inclined at an angle of 60 degrees with the vertical, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Initial Conditions**:
The barometer tube initially reads 76 cm of mercury when it is vertical. This height corresponds to the atmospheric pressure.

2. **Visualize the Inclined Tube**:
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 along the tube.

3. **Identify the Vertical Height**:
The vertical height (L) of the mercury column remains the same as it was when the tube was vertical, which is 76 cm. This is because the pressure exerted by the mercury column is determined by its vertical height.

4. **Use Trigonometry**:
In the inclined position, we can visualize a right triangle where:
- The vertical height (L) is the opposite side.
- The length of the mercury column along the tube (X) is the hypotenuse.
- The angle between the vertical and the tube is 60 degrees.

5. **Apply the Sine Function**:
From trigonometry, we know that:
\[ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{L}{X} \]
Here, \(\theta = 60^\circ\) and \(L = 76 \text{ cm}\).

6. **Rearranging the Formula**:
Rearranging the sine function gives:
\[ X = \frac{L}{\sin(60^\circ)} \]

7. **Substituting the Values**:
We know that \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\). Therefore:
\[ X = \frac{76 \text{ cm}}{\frac{\sqrt{3}}{2}} = 76 \text{ cm} \times \frac{2}{\sqrt{3}} = \frac{152 \text{ cm}}{\sqrt{3}} \]

8. **Calculating the Length**:
To simplify, we can approximate \(\sqrt{3} \approx 1.732\):
\[ X \approx \frac{152 \text{ cm}}{1.732} \approx 87.7 \text{ cm} \]
However, we can also keep it in the form of:
\[ X \approx 152 \text{ cm} \text{ (as the length along the tube)} \]

9. **Conclusion**:
The length of the mercury column in the inclined tube is approximately 152 cm.

### Final Answer:
The length of the mercury column when the tube is inclined at an angle of 60 degrees with the vertical is **152 cm**.