how fast does the level of water in a full 2m high, 10cm diameter tank reduces when a 4cm diameter hole is opened at the bottom?
To find out how fast the level of water in the tank reduces, we need to consider the rate at which water flows out of the tank through the hole. The flow rate can be calculated using Torricelli's law, which states that the flow rate (Q) is equal to the cross-sectional area of the hole (A) multiplied by the velocity of the fluid (v).
To determine the velocity of the fluid, we can use the principle of conservation of energy. The potential energy of the water due to its height will be converted into the kinetic energy of the fluid as it flows out of the hole.
First, let's calculate the cross-sectional area of the hole using the formula for the area of a circle (A = πr²).
Given:
Height of the tank (h) = 2m
Diameter of the tank (D) = 10cm = 0.1m
Radius of the hole (r) = 4cm = 0.04m
Cross-sectional area of the hole (A) = π(0.04)²
= 0.00125664 m²
Now, let's determine the velocity at which water flows out of the hole by considering the potential energy converted to kinetic energy.
Potential energy (PE) = mgh
where m is the mass of the water, g is the acceleration due to gravity, and h is the height of the water column.
We can assume that the mass of the water is proportional to the volume of the water, which is equal to the cross-sectional area multiplied by the height (A × h).
Substituting the values:
PE = (A × h) × g
Velocity (v) is then given by:
v = sqrt(2PE/m)
where m is the mass of the water, which can be approximated to the density (ρ) multiplied by the volume (A × h).
Substituting the values:
v = sqrt(2(A × h) × g / (ρ × (A × h)))
The density of water (ρ) is approximately 1000 kg/m³.
Now, we can determine the velocity (v) at which water flows out of the hole.
v = sqrt(2 × A × h × g / (ρ × A × h))
v = sqrt(2 × g / ρ)
v = sqrt(2 × 9.8 / 1000)
v ≈ 0.198 m/s
Finally, we can calculate the flow rate (Q) using Torricelli's law:
Q = A × v
Q = 0.00125664 m² × 0.198 m/s
Q ≈ 0.0002494 m³/s
Therefore, the level of water in the tank reduces at a rate of approximately 0.0002494 cubic meters per second (m³/s) when the 4cm diameter hole is opened at the bottom.