Concept explainers
(a)
The principal mass moment of inertia at the origin
Answer to Problem B.72P
The principal mass moment of inertia at the origin
The principal mass moment of inertia at the origin
The principal mass moment of inertia at the origin
Explanation of Solution
Given information:
The thin bent plate with uniform density, the weight of both the plates is
The figure below illustrate different centroidal axis.
Figure-(1)
Write the expression for the mass of plate 1.
Here, the weight of both the plates is
Write the expression for the mass of plate 2.
Here, the weight of both the plates is
Write the expression of position vector of the line joining
Here, the vector along the line joining the points
Write the expression for the magnitude of position vector
Here, the magnitude of the position vector
Write the expression for unit vector along
Here, the unit vector along
Write the expression of mass moment of inertia of section 1 about
Here, the mass moment of inertia of section 1 about
Write the expression of mass moment of inertia of section 2 about
Here, the mass moment of inertia of section 2 about
Write the expression of total mass moment of inertia about
Here, the total mass moment of inertia about
Write the expression of mass moment of inertia of section 1 about
Here, the mass moment of inertia of section 1 about
Write the expression of mass moment of inertia of section 2 about
Here, the mass moment of inertia of section 2 about
Write the expression of total mass moment of inertia about
Here, the total mass moment of inertia about
Write the expression of mass moment of inertia of section 1 about
Here, the mass moment of inertia of section 1 about
Write the expression of mass moment of inertia of section 2 about
Here, the mass moment of inertia of section 2 about
Write the expression of total mass moment of inertia about
Here, the total mass moment of inertia about
From, the symmetry in above figure about
Here, the product mass moment of inertia in
From, the symmetry in above figure about
Write the expression for product of mass moment of inertia in
Here, the product mass moment of inertia is
Write the expression for product mass moment of inertia in
Here, the product mass moment of inertia is
Write the expression for product mass moment of inertia in
Here, the product mass moment of inertia in
Write the expression of principal mass moment of inertia
Here, the principal moment of inertia is
Calculation:
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Solving above cubic Equation to get the values of
Conclusion:
The principal mass moment of inertia at the origin
The principal mass moment of inertia at the origin
The principal mass moment of inertia at the origin
(b)
The determine the orientation of principal axis at origin
Answer to Problem B.72P
The orientation of first principal axis at origin
The orientation of first principal axis at origin
The orientation of first principal axis at origin
The orientation of second principal axis at origin
The orientation of second principal axis at origin
The orientation of second principal axis at origin
The orientation of third principal axis at origin
The orientation of third principal axis at origin
The orientation of third principal axis at origin
Explanation of Solution
The figure below represents the orientation of principal axis with respect to the Cartesian coordinates.
Figure-(1)
Write the equation of direction cosines formed by principal axes 1.
Here, the principal mass moment of inertia about principal axis 1 is
Write the expression of orientation of direction cosine about principal axis 1 in
Here, the orientation of direction cosine about principal axis 1 in
Write the expression of orientation of direction cosine about principal axis 2 in
Here, the orientation of direction cosine about principal axis 1 in
Write the expression of orientation of direction cosine about principal axis 1 in
Here, the orientation of direction cosine about principal axis 1 in
Write the equation of direction cosines formed by principal axes 2.
Here, the principal mass moment of inertia about principal axis in
Write the expression of orientation of direction cosine about principal axis 2 in
Here, the orientation of direction cosine about principal axis 2 is
Write the expression of orientation of direction cosine about principal axis 2 in
Here, the orientation of direction cosine about principal axis 2 in
Write the expression of orientation of direction cosine about principal axis 2 in
Here, the orientation of direction cosine about principal axis 2 in
Write the equation of direction cosines formed by principal axes 3.
Here, the principal mass moment of inertia about principal axis 3 is
Write the expression of orientation of direction cosine about principal axis 3 in
Here, the orientation of direction cosine about principal axis 3 in
Write the expression of orientation of direction cosine about principal axis 2 in
Here, the orientation of direction cosine about principal axis 3 in
Write the expression of orientation of direction cosine about principal axis 2 in
Here, the orientation of direction cosine about principal axis 3 in
Calculation:
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Add Equation (XXXIX) and (XL).
Subtract Equation (XXXIX) and (XL).
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Add Equation (XLIV) and(XLV).
Subtract Equation (XLIV) and(XLV).
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Add Equation (XLIX) and (L).
Substract Equation (XLIX) and (L).
Substitute
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Conclusion:
The orientation of first principal axis at origin
The orientation of first principal axis at origin
The orientation of first principal axis at origin
The orientation of second principal axis at origin
The orientation of second principal axis at origin
The orientation of second principal axis at origin
The orientation of third principal axis at origin
The orientation of third principal axis at origin
The orientation of third principal axis at origin
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Chapter B Solutions
Vector Mechanics For Engineers
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