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Chapter 8: Q. 71 (page 203)

If a vertical cylinder of water (or any other liquid) rotates about its axis, as shown in FIGURE CP8.71, the surface forms a smooth curve. Assuming that the water rotates as a unit (i.e., all the water rotates with the same angular velocity), show that the shape of the surface is a parabola described by the equationz=蝇22gr2

Hint: Each particle of water on the surface is subject to only two forces: gravity and the normal force due to the water underneath it. The normal force, as always, acts perpendicular to the surface.

Short Answer

Expert verified

Shape of the parabola is described by the equation z=蝇22gr2

Step by step solution

01

Given information

The figure shows the detail

02

Explanation

First draw Free Body diagram as below

From the diagram equate the vertical and radial force

Gravitational force, F = mg 鈥︹�︹�︹�︹�︹��..(1)

Radial Force Fradial=mr蝇2

Lets consider a very small element at radius dr and the height of the water above the surface is dh

The angle given as

tan蠒=mr蝇2mg=r蝇2g鈥�鈥�鈥�鈥�鈥�.鈥�(2)tan蠒=dzdr鈥�鈥�鈥�鈥�.....鈥�鈥�鈥�(3)

From equation (2) and (3) we get

dzdr=r蝇2gdz=r蝇2gdr

Upon integration we get

z=蝇22gr2

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