A tube of `1 mm` bore is dipped into a vessel containing a liquid of density `0.8 g//cm^(3)`, surface tension `30 "dyne"//"cm"` and angle of contact zero. Calcualte the length which the liquid will occupy in the tube when the tube is held `(a)` vertical `(b)` inclined to the vertical at an angle of `30^(@)`.

Answer

To solve the problem, we need to calculate the height of the liquid column in a tube of 1 mm bore when dipped into a liquid with given properties. We will solve this in two parts: (a) when the tube is held vertically and (b) when it is inclined at an angle of 30 degrees to the vertical.

### Given Data:
- Diameter of the tube (d) = 1 mm = 0.1 cm
- Radius of the tube (r) = d/2 = 0.05 cm
- Density of the liquid (ρ) = 0.8 g/cm³
- Surface tension (T) = 30 dyne/cm
- Angle of contact (θ) = 0 degrees

### Part (a): Tube held vertically

1. **Formula for height of liquid column (H)**:
The height to which the liquid will rise in the tube can be calculated using the formula:
\[ H = \frac{2T \cos \theta}{\rho g r} \]

2. **Substituting values**:
- \( T = 30 \) dyne/cm
- \( \cos(0) = 1 \)
- \( \rho = 0.8 \) g/cm³
- \( g = 980 \) cm/s² (acceleration due to gravity)
- \( r = 0.05 \) cm

Plugging in the values:
\[ H = \frac{2 \times 30 \times 1}{0.8 \times 980 \times 0.05} \]

3. **Calculating**:
- Numerator: \( 2 \times 30 = 60 \)
- Denominator: \( 0.8 \times 980 \times 0.05 = 39.2 \)
\[ H = \frac{60}{39.2} \approx 1.53 \text{ cm} \]

### Part (b): Tube inclined at 30 degrees

1. **Using the relationship between vertical height (H) and length of liquid column (x)**:
When the tube is inclined at an angle of 30 degrees, the relationship is given by:
\[ \cos(30) = \frac{H}{x} \]
Therefore,
\[ x = \frac{H}{\cos(30)} \]

2. **Substituting values**:
- \( H = 1.53 \) cm
- \( \cos(30) = \frac{\sqrt{3}}{2} \)

Plugging in the values:
\[ x = \frac{1.53}{\frac{\sqrt{3}}{2}} = \frac{1.53 \times 2}{\sqrt{3}} \approx \frac{3.06}{1.732} \approx 1.77 \text{ cm} \]

### Final Answers:
- (a) The height of the liquid in the tube when held vertically is approximately **1.53 cm**.
- (b) The length of the liquid column in the tube when inclined at 30 degrees is approximately **1.77 cm**.