   Chapter 15.4, Problem 17E

Chapter
Section
Textbook Problem

Find the moments of inertia Ix, Iy, I0 for the lamina of Exercise 3.

To determine

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

Explanation

Given:

The region D is D={(x,y)|1x3,1y4} .

The density function is ρ(x,y)=ky2 .

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.

If g(x) is the function of x and h(y) is the function of y then,

abcdg(x)h(y)dydx=abg(x)dxcdh(y)dy (1)

Calculation:

The moments of inertia Ix is,

Ix=Dy2ρ(x,y)dA=1314y2ky2dydx=1314ky4dydx

Integrate with respect to x and y by using (1),

Ix=k

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