   Chapter 16.4, Problem 25E

Chapter
Section
Textbook Problem

A plane lamina with constant density ρ(x, y) = ρ occupies region in the xy-plane bounded by a simple closed path C. Show that its moments of inertia about the axes are I x = − ρ 3 ∮ C   y 3 d x I y = ρ 3   ∮ C   x 3 d y

To determine

To show: The moments of inertia of plane lamina about axes as Ix=ρ3Cy3dx and Iy=ρ3Cx3dy .

Explanation

Given data:

The density of plane lamina is ρ(x,y)=ρ .

Formula used:

Consider a region D bounded by a closed path C.

Write the Green’s Theorem in case of sample region D.

CPdx=DPydACQdy=DQxdA

Here,

A is area.

Find the value of ρ3Cy3dx using the Green’s Theorem.

ρ3Cy3dx=ρ3(Dy(y3)dA) {CPdx=DPydA}=ρ3D(3y2)dA {t(tn)=ntn1}=Dy2ρdA=Ix

Find the value of ρ3Cx3dy using the Green’s Theorem

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