To find the velocity of efflux of water from the hole in the cylindrical vessel, we can use Torricelli's theorem. This theorem states that the speed of fluid flowing out of an orifice under the influence of gravity is given by the formula:
Understanding the Variables
Before we dive into the calculations, let's identify the key variables involved:
- Cross-sectional area of the vessel (A): 1000 cm²
- Mass of the piston (m): 10 kg
- Cross-sectional area of the hole (a): 10 mm² (which is 0.001 m²)
- Depth of the hole (h): 50 cm (which is 0.5 m)
- Acceleration due to gravity (g): approximately 9.81 m/s²
Applying Torricelli's Theorem
According to Torricelli's theorem, the velocity of efflux (v) can be calculated using the formula:
v = √(2gh)
Here, h is the height of the water column above the hole, which is 0.5 m in this case. Plugging in the values:
Calculating the Velocity
Now, substituting the known values into the equation:
v = √(2 * 9.81 m/s² * 0.5 m)
Calculating the product inside the square root:
v = √(9.81 m/s²)
Now, taking the square root:
v ≈ 3.13 m/s
Conclusion
The velocity of efflux of water from the hole in the cylindrical vessel is approximately 3.13 m/s. This result illustrates how the height of the water column directly influences the speed at which the water exits the vessel. The greater the height, the higher the velocity, which is a fundamental principle in fluid dynamics.
