# paraboloid

## Partially Filled Cylindrical Tank Rotated at 90 rpm

**Situation**

An open cylindrical vessel 1.3 m in diameter and 2.1 m high is 2/3 full of water. If rotated about the vertical axis at a constant angular speed of 90 rpm,

1. Determine how high is the paraboloid formed of the water surface.

A. 1.26 m | C. 2.46 m |

B. 1.91 m | D. 1.35 m |

2. Determine the amount of water that will be spilled out.

A. 140 L | C. 341 L |

B. 152 L | D. 146 L |

3. What should have been the least height of the vessel so that no water is spilled out?

A. 2.87 m | C. 3.15 m |

B. 2.55 m | D. 2.36 m |

## Rotation - Rotating Vessel

When at rest, the surface of mass of liquid is horizontal at PQ as shown in the figure. When this mass of liquid is rotated about a vertical axis at constant angular velocity ω radian per second, it will assume the surface ABC which is parabolic. Every particle is subjected to centripetal force or centrifugal force CF = mω^{2}x which produces centripetal acceleration towards the center of rotation. Other forces that acts are gravity force W = mg and normal force N.

## Example 1 | Volumes of Solids of Revolution

**Example 1**

Find the volume of the solid generated when the area bounded by the curve y^{2} = x, the x-axis and the line x = 2 is revolved about the x-axis.