A rectangular plate of dimensions `6cm xx 4 cm` and thickness 2 mm is placed with its largest face flat on surface of water . (i) What is the downward force on the plate due to surface tension ? Surface tension of water `=7.0 xx10^(-2) "Nm"^(-1)` (ii) If the plate is placed vertical so that the longest side just touches the water surface, find the downward force on the plate.

Answer

To solve the problem step-by-step, we will break it down into two parts as outlined in the question.

### Part (i): Downward Force on the Plate due to Surface Tension

1. **Identify the dimensions of the plate**:
- Length (L) = 6 cm = 0.06 m
- Width (W) = 4 cm = 0.04 m
- Thickness (T) = 2 mm = 0.002 m (not needed for surface tension calculation)

2. **Calculate the perimeter of the plate that is in contact with the water**:
- Since the plate is placed flat on the water surface, the perimeter (P) that contributes to the surface tension is:
\[ P = 2 \times (L + W) = 2 \times (0.06 + 0.04) = 2 \times 0.10 = 0.20 \text{ m} \]

3. **Use the formula for the force due to surface tension**:
- The force (F) due to surface tension is given by:
\[ F = \text{Surface Tension} \times \text{Perimeter} \]
- Given surface tension (γ) = \(7.0 \times 10^{-2} \text{ Nm}^{-1}\):
\[ F = 7.0 \times 10^{-2} \times 0.20 = 1.4 \times 10^{-2} \text{ N} \]

4. **Conclusion for Part (i)**:
- The downward force on the plate due to surface tension is \(1.4 \times 10^{-2} \text{ N}\).

### Part (ii): Downward Force on the Plate when Vertical

1. **Identify the dimensions of the plate**:
- The plate is now placed vertically, with the longest side (6 cm) touching the water surface.

2. **Calculate the height of the plate that is in contact with the water**:
- In this case, the height (H) that contributes to the surface tension is equal to the thickness of the plate:
\[ H = 0.002 \text{ m} \]

3. **Calculate the perimeter of the plate that is in contact with the water**:
- The perimeter (P) that contributes to the surface tension when the plate is vertical is:
\[ P = 2 \times \text{Height} + \text{Length} = 2 \times 0.002 + 0.06 = 0.004 + 0.06 = 0.064 \text{ m} \]

4. **Use the formula for the force due to surface tension**:
- The force (F) due to surface tension is given by:
\[ F = \text{Surface Tension} \times \text{Perimeter} \]
- Using the same surface tension (γ):
\[ F = 7.0 \times 10^{-2} \times 0.064 = 4.48 \times 10^{-3} \text{ N} \]

5. **Conclusion for Part (ii)**:
- The downward force on the plate when placed vertically is \(4.48 \times 10^{-3} \text{ N}\).

### Summary of the Answers:
- (i) Downward force due to surface tension when flat: \(1.4 \times 10^{-2} \text{ N}\)
- (ii) Downward force due to surface tension when vertical: \(4.48 \times 10^{-3} \text{ N}\)