An open circular tank of 15 cm diameter and 100 cm long contains water up to a height of 80 cm find the
maximum speed at which the cylinder is to be rotated about its vertical axis so that no water spills.
To find the maximum speed at which the cylinder can be rotated so that no water spills out, we need to consider the force acting on the water.
When the cylinder is rotating, the water experiences two forces: gravity and centrifugal force. The centrifugal force acts radially outward and can cause the water to spill out if it exceeds the force of gravity.
The centrifugal force acting on the water can be calculated using the formula:
Centrifugal force (Fc) = m * ω^2 * r
Where:
m = mass of the water
ω = angular velocity (rotational speed) of the cylinder
r = radius of the cylinder
The mass of the water can be calculated using the formula:
mass (m) = density (ρ) * volume (V)
The density of water is a constant value, which is approximately 1000 kg/m^3.
The volume of the water can be calculated using the formula for the volume of a cylinder:
volume (V) = π * r^2 * h
Where:
h = height of the water in the cylinder
Now, let's substitute the values given in the problem:
Diameter of the cylinder = 15 cm = 0.15 m (since 1 cm = 0.01 m)
Radius of the cylinder (r) = 0.075 m (half of the diameter)
Height of the water (h) = 80 cm = 0.8 m (since 1 cm = 0.01 m)
Using these values, we can calculate the mass (m) and volume (V) of the water.
mass (m) = density (ρ) * volume (V)
= 1000 kg/m^3 * (π * 0.075^2 * 0.8) m^3
≈ 141.37 kg
Now, let's find the maximum angular velocity (ω) at which the cylinder can be rotated without spilling water.
Centrifugal force (Fc) = m * ω^2 * r
Since we want to find the maximum speed, we set the centrifugal force equal to the force of gravity acting on the water.
Fc = m * g
Where g is the acceleration due to gravity, which is approximately 9.8 m/s^2.
m * ω^2 * r = m * g
Now we can solve for ω.
ω^2 = g / r
ω = √(g / r)
Substituting the values:
ω = √(9.8 / 0.075) ≈ 16.57 rad/s
So, the maximum speed at which the cylinder can be rotated about its vertical axis so that no water spills is approximately 16.57 rad/s.