   Chapter 15.4, Problem 20E

Chapter
Section
Textbook Problem

Consider a square fan blade with sides of length 2 and the lower left comer placed at the origin. If the density of the blade is ρ (x, y) = 1 + 0.1x, is it more difficult to rotate the blade about the x-axis or the y-axis?

To determine

The direction of the fan blade in which it is more difficult to rotate.

Explanation

Given:

The region D is xy plane occupied by the fan bladewith sides of length 2.

The density function ρ(x,y)=1+0.1x .

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

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

Calculation:

From the given conditions it is observed that x varies from 0 to 2 and y varies from 0 to 2. Obtain the direction of the fan blade in which it is difficult to rotate by find the moments of inertia about x and y axes.

Obtain the moment of inertia Ix .

Ix=Dy2ρ(x,y)dA=0202y2(1+0.1x)dydx=0202(y2+0.1xy2)dydx .

Integrate with respect to y and apply the corresponding limit.

Ix=02(y33+0.1xy33)02dx=02[((2)33+0.1x(2)33)((0)33+0.1x(0)33)]dx=02[(83+0.8x3)(0+0)]dx=02(83+0.8x3)dx

Integrate with respect to x and apply the corresponding limit.

Ix=[8x3+0.8x23(2)]02=[(8(2)3+0

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