   Chapter 15, Problem 41RE

Chapter
Section
Textbook Problem

Consider a lamina that occupies the region D bounded by the parabola x = 1 − y2 and the coordinate axes in the first quadrant with density function ρ(x, y) = y.(a) Find the mass of the lamina.(b) Find the center of mass.(c) Find the moments of inertia and radii of gyration about the x- and y-axes.

(a)

To determine

To find: The mass of the lamina.

Explanation

Given:

The region D is bounded by x=1y2.

The density function is ρ(x,y)=y.

Formula used:

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 center of mass of the lamina that occupies the given region D is (x¯,y¯).

Here, x¯=Mym=1mDxρ(x,y)dA and y¯=Mxm=1mDyρ(x,y)dA

Calculation:

From the given conditions, it is observed that x varies from 0 to 1. Then, the mass of the lamina is,

m=Dρ(x,y)dA=0101y2ydxdy

(b)

To determine

To find: The center of the mass.

(c)

To determine

To find: The moments of inertia and radius of gyration about x and y axis.

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