Chapter 5.P, Problem 12P

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8th Edition

James Stewart

ISBN: 9781285740621

Textbook Problem

A cylindrical container of radius *r* and height *L* is partially filled with a liquid whose volume is *V*. If the container is rotated about its axis of symmetry with constant angular speed

about the

(a) Determine

(b) At what angular speed will the surface of the liquid touch the bottom? At what speed will it spill over the top?

(c) Suppose the radius of the container is 2 ft, the height is 7 ft, and the container and liquid are rotating at the same constant angular speed. The surface of the liquid is 5 ft below the top of the tank at the central axis and 4 ft below the top of the tank 1 ft out from the central axis.

(i) Determine the angular speed of the container and the volume of the fluid.

(ii) How far below the top of the tank is the liquid at the wall of the container?

To determine

**(a)**

Explanation

**1) Concept:**

The Cylindrical Shell method:

The volume of a solid obtained by rotating about the

**2) Given:**

**3) Calculation:**

It is difficult to determine the inner and outer radius to use the washer method. Therefore, use the cylindrical method, and rotation about y axis.

By using cylindrical shell method, the shell has radius x, circumference

To determine

**(b)**

**To find:**

At what speed will the surface of the liquid touch the bottom and the top?

To determine

**(c)**

**To find:**

i) The angular speed of container and volume of the fluid

ii) Liquid at the wall of the container below the top of tank

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