   Chapter 15.4, Problem 21E

Chapter
Section
Textbook Problem

A lamina with constant density ρ (x, y) = ρ occupies the given region. Find the moments of inertia lx and Iy and the radii of gyration x ¯ ¯ and y ¯ ¯ .21. The rectangle 0 ≤ x ≤ b,0 ≤ y ≤ h

To determine

To find: The moments of inertia about x and y axes Ix,Iy and radii of gyration x¯¯,y¯¯ .

Explanation

Given:

The density function, ρ(x,y)=ρ .

The region R is 0xb,0yh .

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)dA

The total mass of the lamina is, m=limk,li=1kj=1lρ(xij*,yij*)ΔA=Dρ(x,y)dA .

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

The radii of gyration about y and x axis is respectively my¯¯2=Ix,mx¯¯2=Iy .

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:

Obtain the moments of inertia Ix .

Ix=Ry2ρ(x,y)dA=0b0hy2ρdydx=ρ0b0hy2dydx

Integrate with respect to x and y using the equation (1).

Ix=ρ0bdx0hy2dy=ρ[x]0b[y33]0h=ρ[b0][(h)33(0)33]=ρ(b)(h33)

= 13ρbh3

Obtain the moments of inertia Iy

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