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 = mω2x which produces centripetal acceleration towards the center of rotation. Other forces that acts are gravity force W = mg and normal force N.
 

008-rotating-vessel.gif

 

Example 1 | Volumes of Solids of Revolution

Example 1
Find the volume of the solid generated when the area bounded by the curve y2 = x, the x-axis and the line x = 2 is revolved about the x-axis.
 

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