A cyclinderical vessel of cross-sectional area `1000cm^(2)`, fitted with a frictonless piston of mass `10kg`, and filled with water completely. A small hole of cross-sectional area `10mm^(2)` is opened at a point `50cm` deep from the lower surface of the piston. The velocity of efflux from the hole will be

Options

A. `10.5m//s`
B. `3.4m//s`
C. `0.8m//s`
D. `0.2m//s`

Answer

To find the velocity of efflux from the hole in the cylindrical vessel, we can use Torricelli's theorem, which states that the speed of efflux of a fluid under the force of gravity through an orifice is given by the equation:

\[ v = \sqrt{2gh} \]

where:
- \( v \) is the velocity of efflux,
- \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \)),
- \( h \) is the height of the fluid column above the hole.

### Step-by-Step Solution:

1. **Identify the parameters:**
- Cross-sectional area of the vessel, \( A = 1000 \, \text{cm}^2 = 0.1 \, \text{m}^2 \)
- Mass of the piston, \( m = 10 \, \text{kg} \)
- Depth of the hole from the lower surface of the piston, \( d = 50 \, \text{cm} = 0.5 \, \text{m} \)
- Density of water, \( \rho = 1000 \, \text{kg/m}^3 \)
- Acceleration due to gravity, \( g = 9.8 \, \text{m/s}^2 \)

2. **Calculate the pressure exerted by the piston:**
\[ F = mg = 10 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 98 \, \text{N} \]
The pressure \( P \) at the depth of the hole is given by:
\[ P = \frac{F}{A} = \frac{98 \, \text{N}}{0.1 \, \text{m}^2} = 980 \, \text{Pa} \]

3. **Calculate the height of the water column above the hole:**
The total height of the water column \( H \) is the height of the piston plus the height of the water above the hole. The height of the water column above the hole is:
\[ h = H - d \]
To find \( H \), we can use the relationship between pressure and height:
\[ P = \rho g h \implies h = \frac{P}{\rho g} \]
Substituting the values:
\[ h = \frac{980 \, \text{Pa}}{1000 \, \text{kg/m}^3 \times 9.8 \, \text{m/s}^2} = 0.1 \, \text{m} = 10 \, \text{cm} \]
Therefore, the height of the water column above the hole is:
\[ H = d + h = 0.5 \, \text{m} + 0.1 \, \text{m} = 0.6 \, \text{m} \]

4. **Calculate the velocity of efflux:**
Using Torricelli's theorem:
\[ v = \sqrt{2gh} = \sqrt{2 \times 9.8 \, \text{m/s}^2 \times 0.6 \, \text{m}} = \sqrt{11.76} \approx 3.43 \, \text{m/s} \]

5. **Final result:**
The velocity of efflux from the hole is approximately \( 3.4 \, \text{m/s} \).

### Conclusion:
The velocity of efflux from the hole will be \( 3.4 \, \text{m/s} \).