Answered: Density = 2 a a CF, J.7) Figure 16.75

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Question

Center of mass with variable density Find the center of mass of the
interior of the hemisphere D of radius a with its base on the xy-plane. The density of the object given in spherical coordinates is ƒ(ρ, φ, θ) = 2 - ρ/a (heavy near the center and light near the outer surface; as shown).

Expert Solution

Step 1

Given Data:

The density of the hemisphere is given as $d = 2 - \frac{ρ}{a}$

To determine the centre of mass of the hemisphere, take a hemispherical shell of radius $ρ$ and thickness $d ρ$ . The centre of mass of this spherical shell has coordinate $(0, 0, \frac{ρ}{2})$ as shown in the figure.

Step 2

The mass dm of the spherical shell can be determined as,

$\begin{array}{rcl} d m & = & d (d V) \\ = & (2 - \frac{ρ}{a}) (2 π ρ^{2} d ρ) \end{array}$

The expression for the location of the centre of mass is,

$x = \frac{\int x d m}{\int d m} y = \frac{\int y d m}{\int d m} z = \frac{\int z d m}{\int d m}$

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