A cylindrical vessel of height 500mm has an orifice (small hole) at its bottom. The orifice is initially closed and water is filled in it up to height H. Now the top is completely sealed with a cap and the orifice at the bottom is opened. Some water comes out from the orifice and the water level in the vessel becomes steady with height of water column being 200mm. Find the fall in height(in mm) of water level due to opening of the orifice.
[Take atmospheric pressure `=1.0xx10^5N//m^2`, density of water=1000kg//m^3` and `g=10m//s^2`. Neglect any effect of surface tension.]

Options

A. 6
B. 4
C. 2
D. 8

Explanation

To solve the problem step by step, we will analyze the situation involving the cylindrical vessel and the orifice at the bottom.

### Step 1: Understand the Initial Conditions
The cylindrical vessel has a height of 500 mm, and it is filled with water up to an unknown height \( H \). When the orifice at the bottom is opened, the water level drops and stabilizes at 200 mm.

### Step 2: Identify the Pressure at the Water Level
When the water level is at 200 mm, we can calculate the pressure at that height using the hydrostatic pressure formula:
\[ P = P_0 - \rho g h \]
Where:
- \( P_0 = 1.0 \times 10^5 \, \text{N/m}^2 \) (atmospheric pressure)
- \( \rho = 1000 \, \text{kg/m}^3 \) (density of water)
- \( g = 10 \, \text{m/s}^2 \) (acceleration due to gravity)
- \( h = 0.2 \, \text{m} \) (height of water column when stabilized)

Substituting the values:
\[ P = 1.0 \times 10^5 - (1000 \times 10 \times 0.2) \]
\[ P = 1.0 \times 10^5 - 2000 = 98000 \, \text{N/m}^2 \]

### Step 3: Relate the Pressure to the Original Height
The pressure at the original height \( H \) before the orifice was opened can be expressed as:
\[ P_0 = P + \rho g (H - 0.2) \]
Substituting the known values:
\[ 1.0 \times 10^5 = 98000 + 1000 \times 10 \times (H - 0.2) \]
\[ 1.0 \times 10^5 - 98000 = 1000 \times 10 \times (H - 0.2) \]
\[ 2000 = 10000 (H - 0.2) \]
\[ H - 0.2 = \frac{2000}{10000} = 0.2 \]
\[ H = 0.4 \, \text{m} = 400 \, \text{mm} \]

### Step 4: Calculate the Fall in Height
The fall in height of the water level due to the opening of the orifice is given by:
\[ \text{Fall in height} = H - \text{final height} \]
\[ \text{Fall in height} = 400 \, \text{mm} - 200 \, \text{mm} = 200 \, \text{mm} \]

### Final Result
The fall in height of the water level due to the opening of the orifice is **200 mm**.

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