International Edition---engineering Mechanics: Statics, 4th Edition
International Edition---engineering Mechanics: Statics, 4th Edition
4th Edition
ISBN: 9781305501607
Author: Andrew Pytel And Jaan Kiusalaas
Publisher: CENGAGE L
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Chapter 9, Problem 9.51P
To determine

(a)

The principal moments of inertia and the principal directions at the centroid C for the semicircular region.

Expert Solution
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Answer to Problem 9.51P

The principal moments of inertia:

  I1=81.43×106 mm4

  I2=22.77×106 mm4

Principal directions are along x and y axes

Explanation of Solution

Given information:

The semicircular region:

International Edition---engineering Mechanics: Statics, 4th Edition, Chapter 9, Problem 9.51P , additional homework tip  1

Calculations:

International Edition---engineering Mechanics: Statics, 4th Edition, Chapter 9, Problem 9.51P , additional homework tip  2

Because to symmetry, the x- and y- axes are the principal axes at C.

Hence,

  I1=Iy=π8R4=π8(120)4I1=81.43×106 mm4I2=Ix=0.1098R4=0.1098(120)4I2=22.77×106 mm4

Conclusion:

The principal moments of inertia at the centroid C for the semicircular region shown are I1=81.43×106 mm4 and I2=22.77×106 mm4. And the principal directions are along x and y axes.

To determine

(b)

The moments and the products of inertia about the u-v-axes for the semicircular region shown.

Expert Solution
Check Mark

Answer to Problem 9.51P

Moments of inertia:

  Iu=33.2×106mm4

  Iv=71.0×106mm4

Products of inertia:

  Iuv=22.5×106mm4

Explanation of Solution

Given information:

For the semicircular region shown:

  I1=Iy=81.43×106 mm4

  I2=Ix=22.77×106 mm4

Calculations:

  12(Ix+Iy)=12(22.77+81.43)×106=52.10×106 mm412(IxIy)=12(22.7781.43)×106=29.33×106 mm4Moments of inertia about the u-v-axes, using the relations:Iu=12(Ix+Iy)+12(IxIy)cos2θIxysin2θIu=[52.1029.33 cos( 50 o)0]×106Iu=33.2×106mm4Iv=12(Ix+Iy)12(IxIy)cos2θ+Ixysin2θIv=[52.10+29.33 cos( 50 o)+0]×106Iv=71.0×106mm4Hence, Products of inertia about the u-v-axes:Iuv=12(IxIy)sin2θ+Ixycos2θIuv=[29.33 sin( 50 o)+0]×106Iuv=22.5×106mm4

Conclusion:

For the semicircular region, the moments of inertia about the u-v axes are Iu=33.2×106mm4 and Iv=71.0×106mm4. And the products of inertia about the u-v axes is Iuv=22.5×106mm4.

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Chapter 9 Solutions

International Edition---engineering Mechanics: Statics, 4th Edition

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