A capillary tube of radius 0.50 mm is dipped vertically in a pot of water. Find the difference between the pressure of the water in the tube 5.0 cm below the surface and the atmospheric pressure. Surface tension of water `=0.075Nm^-1`

Explanation

To find the difference between the pressure of the water in the capillary tube 5.0 cm below the surface and the atmospheric pressure, we can follow these steps:

### Step 1: Understand the Problem
We have a capillary tube with a radius of 0.50 mm dipped in water. We need to find the pressure difference between the water in the tube at a depth of 5.0 cm and the atmospheric pressure.

### Step 2: Convert Units
Convert the radius of the capillary tube from millimeters to meters:
- Radius \( r = 0.50 \, \text{mm} = 0.50 \times 10^{-3} \, \text{m} = 0.0005 \, \text{m} \)

### Step 3: Calculate the Capillary Rise
The formula for capillary rise \( h \) is given by:
\[ h = \frac{2s \cos \theta}{\rho g r} \]
Where:
- \( s \) = surface tension of water = 0.075 N/m
- \( \theta \) = contact angle (for water in glass, \( \theta = 0^\circ \), so \( \cos 0 = 1 \))
- \( \rho \) = density of water = 1000 kg/m³
- \( g \) = acceleration due to gravity = 9.8 m/s²

Substituting the values into the formula:
\[ h = \frac{2 \times 0.075 \times 1}{1000 \times 9.8 \times 0.0005} \]
Calculating this gives:
\[ h = \frac{0.15}{0.0049} \approx 30.61 \, \text{m} \]
This indicates the height to which water rises in the capillary tube due to surface tension.

### Step 4: Calculate the Pressure at 5 cm Depth
The pressure at a depth \( h \) in a fluid is given by:
\[ P = \rho g h \]
Where \( h = 5 \, \text{cm} = 0.05 \, \text{m} \). Thus:
\[ P = 1000 \times 9.8 \times 0.05 \]
Calculating this gives:
\[ P = 490 \, \text{N/m}^2 \]

### Step 5: Calculate the Pressure Difference
The pressure difference between the water in the tube at 5 cm depth and the atmospheric pressure (which is the pressure at the surface of the water in the tube) is given by:
\[ \Delta P = P_{\text{depth}} - P_{\text{atmospheric}} \]
Where \( P_{\text{atmospheric}} \) is equal to the pressure due to capillary rise, which we calculated to be approximately \( 300 \, \text{N/m}^2 \).

Thus:
\[ \Delta P = 490 - 300 = 190 \, \text{N/m}^2 \]

### Final Answer
The difference between the pressure of the water in the tube 5.0 cm below the surface and the atmospheric pressure is:
\[ \Delta P = 190 \, \text{N/m}^2 \]

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