To solve the problem of determining the radii of curvature of the upper and lower meniscuses in a capillary tube containing a drop of water, we can follow these steps:
### Step 1: Understand the Setup
We have a capillary tube with an internal diameter of 1 mm, which means the radius \( R \) of the tube is:
\[
R = \frac{1 \, \text{mm}}{2} = 0.5 \, \text{mm} = 0.5 \times 10^{-3} \, \text{m}
\]
### Step 2: Calculate the Pressure due to Surface Tension for the Upper Meniscus
The pressure difference due to the surface tension at the upper meniscus (which is concave downwards) is given by:
\[
P_s = \frac{2T}{R_1}
\]
where \( T \) is the surface tension of water, \( T = 0.073 \, \text{N/m} \).
### Step 3: Set the Pressure Equal to Hydrostatic Pressure
When wetting is complete, the pressure \( P_1 \) at the upper meniscus can be equated to the pressure due to surface tension:
\[
P_1 = P_s
\]
Thus, we have:
\[
P_1 = \frac{2T}{R_1}
\]
Since \( R = 0.5 \, \text{mm} \) is the radius of the capillary tube, we can substitute \( R_1 = R \):
\[
P_1 = \frac{2 \times 0.073}{0.5 \times 10^{-3}} = 292 \, \text{N/m}^2
\]
### Step 4: Calculate the Hydrostatic Pressure at the Lower Meniscus
The drop forms a column of water with a height of 4 cm (or 0.04 m). The hydrostatic pressure \( P_2 \) at the lower meniscus is given by:
\[
P_2 = \rho h g
\]
where \( \rho \) (density of water) is \( 1000 \, \text{kg/m}^3 \), \( h = 0.04 \, \text{m} \), and \( g = 9.8 \, \text{m/s}^2 \):
\[
P_2 = 1000 \times 0.04 \times 9.8 = 392 \, \text{N/m}^2
\]
### Step 5: Determine the Radius of Curvature for the Lower Meniscus
Since \( P_2 > P_1 \), the lower meniscus is concave upwards. The pressure difference can be expressed as:
\[
P_2 - P_1 = \frac{2T}{R_2}
\]
Substituting the values we calculated:
\[
392 - 292 = \frac{2 \times 0.073}{R_2}
\]
This simplifies to:
\[
100 = \frac{0.146}{R_2}
\]
Rearranging gives:
\[
R_2 = \frac{0.146}{100} = 0.00146 \, \text{m} = 1.46 \, \text{mm}
\]
### Final Results
- The radius of curvature of the upper meniscus \( R_1 \) is \( 0.5 \, \text{mm} \).
- The radius of curvature of the lower meniscus \( R_2 \) is \( 1.46 \, \text{mm} \).
### Summary
- \( R_1 = 0.5 \, \text{mm} \)
- \( R_2 = 1.46 \, \text{mm} \)
