A barometer tube is inclined at an angle of `30^@` with horizontal direction. Calculate length of mercury column in tube if atmospheric pressure is 76 cm of mercury.

Answer

To solve the problem of finding the length of the mercury column in a barometer tube inclined at an angle of \(30^\circ\) with the horizontal, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship Between Length and Height**:
The atmospheric pressure is given as the height of the mercury column when the tube is vertical. When the tube is inclined, the effective height of the mercury column (h) can be related to the actual length of the mercury column (L) using trigonometric functions.

2. **Identify the Known Values**:
- Atmospheric pressure (height of mercury column when vertical), \(h = 76 \, \text{cm}\).
- Angle of inclination, \(\theta = 30^\circ\).

3. **Use the Cosine Function**:
When the tube is inclined, the vertical height \(h\) can be expressed in terms of the length \(L\) of the mercury column:
\[ h = L \cos(\theta) \]
Rearranging this gives:
\[ L = \frac{h}{\cos(\theta)} \]

4. **Calculate the Cosine of the Angle**:
For \(\theta = 30^\circ\):
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \approx 0.866 \]

5. **Substitute the Values**:
Now, substituting the known values into the equation:
\[ L = \frac{76 \, \text{cm}}{\cos(30^\circ)} = \frac{76 \, \text{cm}}{0.866} \]

6. **Perform the Calculation**:
\[ L \approx \frac{76 \, \text{cm}}{0.866} \approx 87.7 \, \text{cm} \]

7. **Final Result**:
The length of the mercury column in the inclined tube is approximately \(87.7 \, \text{cm}\).