Concept explainers
(a)
The principal mass moment of inertias at the origin.
Answer to Problem B.74P
The principal mass moment of inertias at the origin are
Explanation of Solution
Given information:
The mass per unit length of the steel is
Draw the diagram for the different section of the body.
Figure-(1)
Concept used:
Write the expression for the mass of each section.
Here, the mass per unit length is
Write the expression of mass moment of inertia of section 1 about
Write the expression of mass moment of inertia of section 2 about
Write the expression of total mass moment of inertia about
Here, the mass moment of inertia for section 3 about x- axis is
The mass moment of inertia for section 1 is equal to the mass moment of inertia for section 3.
The mass moment of inertia for section 1 is equal to the mass moment of inertia for section 4.
The mass moment of inertia for section 1 is equal to the mass moment of inertia for section 6.
The mass moment of inertia for section 2 is equal to the mass moment of inertia for section 5.
Substitute
Write the expression of total mass moment of inertia about
Here, the mass moment of inertia for section 3 about y- axis is
The mass moment of inertia for section 1 is equal to zero.
The mass moment of inertia for section 4 is equal to the mass moment of inertia for section 5.
The mass moment of inertia for section 2 is equal to the mass moment of inertia for section 6.
Substitute
Write the expression of mass moment of inertia of section 2 about
Write the expression of mass moment of inertia of section 3 about
Write the expression of mass moment of inertia of section 4 about
The figure below illustrates the centroidal axis of a component.
Figure-(2)
From the symmetry in above figure about
Here, the product mass moment of inertia in
From the symmetry in the above figure about
Here, the product mass moment of inertia in
From the symmetry in the above figure about
Here, the product mass moment of inertia in
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 mass moment of inertia with respect o origin along the unit vector
Here, the mass moment of inertia with respect to origin along unit vector
Calculation:
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
The mass moment of inertia about z- axis and the mass moment of inertia about y- axis is equal due to symmetry.
Hence,
Substitute
Substitute
Substitute
After solving the above equation,
Conclusion:
The principal mass moment of inertias at the origin are
(b)
The principal axis about the origin.
Answer to Problem B.74P
The principal axis about the origin are
Explanation of Solution
Given information:
The mass per unit length of the steel is
Draw the diagram for the different section of the body.
Figure-(1)
Calculation:
The direction cosine is calculated as follows:
Substitute the values from the sub-part (a) in above equations as follows:
After solving above equations,
Direction cosine in x direction is calculated as follows:
Direction cosine in y and z direction are,
So, the direction is calculated as follows:
Again,
The direction cosine is calculated as follows:
Substitute the values from the sub-part (a) in above equations as follows:
After solving above equations,
Direction cosine in x direction is calculated as follows:
Direction cosine in z direction is,
So, the direction is calculated as follows:
Similarly,
The direction cosine is calculated as follows:
Substitute the values from the sub-part (a) in above equations as follows:
After solving above equations,
Direction cosine in x direction is calculated as follows:
Direction cosine in y and z direction are,
So, the direction is calculated as follows:
The sketch is shown below:
Conclusion:
So, the principal axis about the origin are
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Chapter B Solutions
Vector Mechanics For Engineers
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