A glass plate of length `10 cm`, breadth `1.54 cm` and thickness `0.20 cm` weigh `8.2 g` in air. It is held vertically with the long side horizontal and the lower half under water. Find the apparent weight of the plate. Surface tension of water `=7.3xx10^(-2)N//m` and `=9.8ms^(-12)`

Explanation

To find the apparent weight of the glass plate when it is held vertically with the lower half submerged in water, we will follow these steps:

### Step 1: Calculate the volume of the submerged portion of the plate
The volume of the submerged portion of the plate can be calculated using the formula for the volume of a rectangular prism:

\[ \text{Volume} = \text{Length} \times \text{Breadth} \times \text{Height} \]

Given:
- Length = 10 cm = 0.10 m
- Breadth = 1.54 cm = 0.0154 m
- Height of submerged portion = half of thickness = 0.20 cm / 2 = 0.10 cm = 0.001 m

Now, substituting the values:

\[ \text{Volume} = 0.10 \, \text{m} \times 0.0154 \, \text{m} \times 0.001 \, \text{m} = 0.00000154 \, \text{m}^3 \]

### Step 2: Calculate the weight of the water displaced
Using the density of water (approximately \(1000 \, \text{kg/m}^3\)), we can find the weight of the water displaced:

\[ \text{Weight of water displaced} = \text{Volume} \times \text{Density} \times g \]

Substituting the values:

\[ \text{Weight of water displaced} = 0.00000154 \, \text{m}^3 \times 1000 \, \text{kg/m}^3 \times 9.8 \, \text{m/s}^2 = 0.015092 \, \text{N} \]

### Step 3: Calculate the downward force due to surface tension
The downward force due to surface tension can be calculated as:

\[ \text{Force due to surface tension} = \text{Surface tension} \times \text{Length in contact with water} \]

The length in contact with water is the total length of the plate, which is:

\[ \text{Total length} = 2 \times \text{Length} + \text{Thickness} = 2 \times 0.10 \, \text{m} + 0.002 \, \text{m} = 0.204 \, \text{m} \]

Now substituting the values:

\[ \text{Force due to surface tension} = 0.073 \, \text{N/m} \times 0.204 \, \text{m} = 0.014892 \, \text{N} \]

### Step 4: Calculate the resultant upthrust
The resultant upthrust can be calculated as:

\[ \text{Resultant upthrust} = \text{Weight of water displaced} - \text{Force due to surface tension} \]

Substituting the values:

\[ \text{Resultant upthrust} = 0.015092 \, \text{N} - 0.014892 \, \text{N} = 0.0002 \, \text{N} \]

### Step 5: Calculate the apparent weight of the plate
The apparent weight of the plate can be calculated as:

\[ \text{Apparent weight} = \text{Weight of the plate in air} - \text{Resultant upthrust} \]

Given the weight of the plate in air is \(8.2 \, \text{g} = 0.0082 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 0.08056 \, \text{N}\).

Now substituting the values:

\[ \text{Apparent weight} = 0.08056 \, \text{N} - 0.0002 \, \text{N} = 0.08036 \, \text{N} \]

### Step 6: Convert the apparent weight back to grams
To convert the apparent weight back to grams:

\[ \text{Apparent weight in grams} = \frac{0.08036 \, \text{N}}{9.8 \, \text{m/s}^2} \approx 8.19 \, \text{g} \]

### Final Answer
The apparent weight of the plate when submerged is approximately **8.19 g**.

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