   Chapter 15.4, Problem 18E

Chapter
Section
Textbook Problem

Find the moments of inertia Ix, Iy, I0 for the lamina of Exercise 6.

To determine

To find: The moments of inertia Ix,Iy,I0 .

Explanation

Given:

The region D is the triangle enclosed by the lines y=0,y=2x,x+2y=1

The density function is ρ(x,y)=x .

Formula used:

The moments of inertia is,

Ix=limm,ni=1mj=1n(yij*)2ρ(xij*,yij*)ΔA=Dy2ρ(x,y)dAIy=limm,ni=1mj=1n(xij*)2ρ(xij*,yij*)ΔA=Dx2ρ(x,y)dAI0=Ix+Iy

Here, the density function is given by ρ(x,y) and D  is the region that is occupied by the lamina.

Calculation:

Obtain the moment of inertia Ix .

Ix=Dy2ρ(x,y)dA=025y212yy2xdxdy .

Integrate with respect to x and apply it’s limit.

Ix=025(y2x22)y212ydy=025[y2(12y)22y2(y2)22]dy=025[y224y32+4y42y42(4)]dy=025(y222y3+15y48)dy

Integrate with respect to y and apply the limit.

Ix=[y32(3)2y44+15y58(5)]025=((25)36(25)42+15(25)540)((0)36(0)42+15(0)540)=((8)6(125)(16)2(625)+15(32)40(3125))(00+0)=8750161250+480125000

= 169375

Obtain the moment of inertia Iy

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