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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

A section of sheet steel 2 mm thick is cut and bent into the machine component shown. Knowing that the density of steel is 7850 kg/m3 , determine the mass products of inertia Ixy, Iyz, and Izx of the component.

Expert Solution

Step 1

Given,

thickness t = 2 mm

density ρ = 7850 kg/m³

Step 2

Mass of both section,

$m_{1} = ρ V_{1} k g m_{1} = ρ (A_{1} t) k g m_{1} = 7850 (0.45 X 0.4) (0.002) k g m_{1} = 2.826 k g m_{2} = ρ V_{2} k g m_{2} = ρ (A_{2} t) k g m_{2} = 7850 (\frac{1}{2} X 0.18 X 0.45) (0.002) k g m_{2} = 0.636 k g$

Step 3

Now from above figure we can say that,

For Section 1,

$(I$ _x'y')1=(I_y'z')1=(Iz'x')1=0

For section 2,

${(I_{y' z'})}_{2} = {(I_{z' x'})}_{2} = 0$

Step 4

${(I_{x' y'})}_{2} = I_{C G} + A (\bar{x} \bar{y}) m^{4} {(I_{x' y'})}_{2} = (- \frac{1}{24} b^{2} h^{2}) + (\frac{1}{2} b h) (\frac{b}{3} X \frac{h}{6}) m^{4} {(I_{x' y'})}_{2} = (- \frac{1}{72} b^{2} h^{2}) m^{4} - P r o d u c t o f M o m e n t o f I n e r t i a o f a r e a$

Product of Moment of Inertia for mass,

${(I_{x' y'})}_{2} = ρ t (- \frac{1}{72} b^{2} h^{2}) {(I_{x' y'})}_{2} = \frac{m_{2}}{A_{2}} (- \frac{1}{36} b h X A_{2}) {(I_{x' y'})}_{2} = - \frac{1}{36} m_{2} b h k g m^{2} . . . . . (1)$

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