If a 5.0 cm long capillary tube with 0.10 mm internal diameter open at both end is slightly dipped in water having surface tension 75 dyne ` cm ^( -1) ` Find radius of curvature of meniscus at top in mm.

Explanation

To find the radius of curvature of the meniscus at the top of a capillary tube dipped in water, we can follow these steps:

### Step 1: Understand the given data
- Length of the capillary tube (L) = 5.0 cm
- Internal diameter of the tube (d) = 0.10 mm = 0.01 cm
- Surface tension of water (T) = 75 dyne/cm = 75 × 10^-5 N/cm (since 1 dyne = 10^-5 N)
- Density of water (ρ) = 1 g/cm³ = 1000 kg/m³
- Acceleration due to gravity (g) = 980 cm/s²
- Contact angle (θ) = 0° (for water in a glass tube)

### Step 2: Calculate the radius of the capillary tube
The radius (r) of the capillary tube is half of the diameter:
\[ r = \frac{d}{2} = \frac{0.10 \text{ mm}}{2} = 0.05 \text{ mm} = 0.005 \text{ cm} \]

### Step 3: Use the formula for capillary rise
The height of the liquid column in a capillary tube is given by:
\[ h = \frac{2T \cos \theta}{r \rho g} \]

Substituting the known values:
- \( T = 75 \text{ dyne/cm} = 75 \times 10^{-5} \text{ N/cm} \)
- \( \cos(0°) = 1 \)
- \( r = 0.005 \text{ cm} \)
- \( \rho = 1 \text{ g/cm}^3 = 1000 \text{ kg/m}^3 \)
- \( g = 980 \text{ cm/s}^2 \)

### Step 4: Calculate the height (h)
Substituting the values into the formula:
\[ h = \frac{2 \times 75 \times 10^{-5} \times 1}{0.005 \times 1 \times 980} \]
\[ h = \frac{150 \times 10^{-5}}{0.005 \times 980} \]
\[ h = \frac{150 \times 10^{-5}}{4.9} \]
\[ h \approx 30.61 \text{ cm} \]

### Step 5: Relate the radius of curvature (R) to the height (h)
The relationship between the radius of curvature (R) and the height (h) in the capillary tube can be expressed as:
\[ R \cdot h = \text{constant} \]

### Step 6: Set up the equation for the radius of curvature
Since the maximum height of the liquid column is greater than the length of the capillary tube, we can express:
\[ R = \frac{r \cdot h_{max}}{h_{actual}} \]
Where:
- \( h_{max} = 30.61 \text{ cm} \)
- \( h_{actual} = 5.0 \text{ cm} \)

### Step 7: Substitute the values to find R
\[ R = \frac{0.005 \text{ cm} \cdot 30.61 \text{ cm}}{5.0 \text{ cm}} \]
\[ R = \frac{0.15305 \text{ cm}}{5.0} \]
\[ R = 0.03061 \text{ cm} \]

### Step 8: Convert R to mm
To convert cm to mm:
\[ R = 0.03061 \text{ cm} \times 10 = 0.3061 \text{ mm} \]

### Final Answer
The radius of curvature of the meniscus at the top is approximately **0.31 mm**.

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