Question

An open cylindrical tank of 2 m diameter and 4 m high, contains water up to 1.5 m depth. If the cylinder rotates about the vertical axis, what angular velocity can be attained without spilling any water?

1. 12.9 radians/sec
2. 10.9 radians/sec
3. 9.9 radians/sec
4. 11.1 radians/sec

Answer 1

Correct Answer - Option 3 : 9.9 radians/sec

Concept:

For no spilling of water, rise above the original water level is equal to fall below the original water level

Height of paraboloid of revolution, \(H = \frac{{{ω ^2}{R^2}}}{{2g}}\)

where, 

ω = Angular velocity

R = Radius of cylinder

Calculations:

Given Data:

Diameter = 2 m

Height of Tank = 4 m

Depth of water = 1.5 m

So, Radius, R = 1 m

Rise above the original water level for no spilling = 4 - 1.5 = 2.5 m

As, Rise above level = Fall below level

So, Fall = 2.5 m

So, the total height of paraboloid = Rise + Fall = 2.5 + 2.5 = 5 m

Putting all that data in above equation, we get,

\(5 = \frac{{{ω ^2} \times {1^2}}}{{2 \times 9.81}}\)

ω = 9.9 rad/sec

Note: 

Although the total height of the cylinder is 4 m, an imaginary paraboloid will be assumed which will be extending below the bottom level of the cylinder up to 1 m, thus, making the total height of the paraboloid 5 m.