Vector Mechanics for Engineers: Statics and Dynamics
Vector Mechanics for Engineers: Statics and Dynamics
12th Edition
ISBN: 9781259638091
Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self
Publisher: McGraw-Hill Education
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Chapter 9, Problem 9.185RP

Determine by direct integration the moments of inertia of the shaded area with respect to the x and y axes.

Chapter 9, Problem 9.185RP, Determine by direct integration the moments of inertia of the shaded area with respect to the x and

Fig. P9.185

Expert Solution & Answer
Check Mark
To determine

Find the moment of inertia of the shaded area with respect to x and y axes.

Answer to Problem 9.185RP

The moment of inertia of the shaded area with respect to x axes is 16105ah3_.

The moment of inertia of the shaded area with respect to y axes is 15ha3_

Explanation of Solution

Given information:

The curve Equation is y=4h(xax2a2)

Calculation:

Sketch the shaded portion with vertical strip as shown in Figure 1.

Vector Mechanics for Engineers: Statics and Dynamics, Chapter 9, Problem 9.185RP

Refer to Figure 1.

Write the curve Equation as shown below:

y=4h(xax2a2) (1)

Determine the moment of inertia (dIx) with respect to x axis of a rectangular strip:

dIx=13y3dx (2)

Substitute 4h(xax2a2) for y in Equation (2).

dIx=13y3dx=13[4h(xax2a2)]3dx (3)

Integrate Equation (3) with respect to x.

Ix=dIx=64h330a(x3a33(x4a4)+3(x5a5)x6a6)=64h33[(x44a33(x55a4)+3(x66a5)x77a6)]0a=64h33[(14x4a335(x5a4)+12(x6a5)17x7a6)]0a

=64h33[(14a4a335(a5a4)+12(a6a5)17a7a6)]=64h33a(1435+1217)=64h33a(1140)Ix=16105ah3

Thus, the moment of inertia of the shaded area with respect to x axes is 16105ah3_.

Determine the area of the strip element dA as shown in below:

dA=ydx

Determine the moment of inertia (dIy) with respect to y axis of a rectangular strip:

dIy=x2dA=4hx2(xax2a2)dx (4)

Integrate Equation (4) with respect to y.

Iy=4h0a(x3ax4a2)dx=4h[x44ax55a2]0a=4h[a44aa55a2]=4h[a34a35]

=4ha3[1415]=4ha3[120]Iy=15ha3

Thus, the moment of inertia of the shaded area with respect to y axes is 15ha3_.

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Chapter 9 Solutions

Vector Mechanics for Engineers: Statics and Dynamics

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