The lower end of a capillary tube is at a depth of `12 cm` and water rises `3 cm` in it. The mouth pressure required to blow an air bubble at the lower end will be `x cm` of water column, where `x` is

Options

A. `12`
B. `15`
C. `3`
D. `9`

Explanation

To solve the problem, we need to determine the pressure required to blow an air bubble at the lower end of the capillary tube. We can follow these steps:

### Step 1: Understand the situation
The lower end of the capillary tube is at a depth of 12 cm below the water level, and water rises 3 cm in the tube. Therefore, the effective depth of the water column above the air bubble will be the depth of the tube minus the height to which the water has risen.

### Step 2: Calculate the effective depth
The effective depth (h) of the water column above the air bubble can be calculated as:
\[ h = \text{Total depth} - \text{Height of water rise} \]
Substituting the values:
\[ h = 12 \, \text{cm} - 3 \, \text{cm} = 9 \, \text{cm} \]

### Step 3: Relate pressure to the height of the water column
The pressure required to blow an air bubble at the lower end of the capillary tube is equal to the pressure exerted by the water column above it. This pressure can be expressed in terms of the height of the water column:
\[ P = \rho g h \]
Where:
- \( P \) is the pressure,
- \( \rho \) is the density of water (approximately \( 1000 \, \text{kg/m}^3 \)),
- \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)),
- \( h \) is the height of the water column in meters.

### Step 4: Convert height to meters
Convert the height from centimeters to meters:
\[ h = 9 \, \text{cm} = 0.09 \, \text{m} \]

### Step 5: Calculate the pressure
Substituting the values into the pressure equation:
\[ P = 1000 \, \text{kg/m}^3 \times 9.81 \, \text{m/s}^2 \times 0.09 \, \text{m} \]
Calculating this gives:
\[ P = 1000 \times 9.81 \times 0.09 = 882.9 \, \text{Pa} \]

### Step 6: Convert pressure to cm of water column
To convert pressure in Pascals to height of water column in cm, we use the relation:
\[ P = \rho g h \Rightarrow h = \frac{P}{\rho g} \]
Substituting \( P = 882.9 \, \text{Pa} \):
\[ h = \frac{882.9}{1000 \times 9.81} \approx 9 \, \text{cm} \]

### Final Result
Therefore, the mouth pressure required to blow an air bubble at the lower end of the capillary tube will be approximately \( 9 \, \text{cm} \) of water column.

### Summary
The value of \( x \) is \( 9 \, \text{cm} \).

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