   Chapter 15.4, Problem 22E

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 ¯ ¯ .22. The triangle with vertices (0, 0), (b, 0), and (0, 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 D is the triangle with vertices (0,0),(b,0),(0,h) .

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 .

Calculation:

Obtain the equation of the line joining the points (b,0) and (0,h) .

xbyh=1yh=1xby=hhxb

From the given conditions, it is observed that, x varies from 0 to b and y varies from 0 to hhxb .

Obtain the moments of inertia Ix .

Ix=Ry2ρ(x,y)dA=0b0hhxby2ρdydx=ρ0b0hhxby2dydx

Integrate with respect to y.

Ix=ρ0b[y33]0hhxbdx=ρ0b[(hhxb)33(0)33]dx=ρ30b[(hhxb)30]dx=ρ30b(hhxb)3dx

Integrate with respect to x.

Ix=ρ3[(hhxb)44(hb)]0b=ρ3(14(hb))[(hhbb)4(hh(0)b)4]=ρ3(b4h)[(0)4(h0)4]=ρb12h[h4]

= ρbh312

Obtain the moments of inertia Iy

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