Concept explainers
The mass moment of inertia of the machine component with respect to the axis through the origin characterized by the unit
Answer to Problem B.61P
The mass moment of inertia of the machine component with respect to the axis through the origin by the unit vector is
Explanation of Solution
Given information:
The diameter of the formed steel wire is
The below figure represents the schematic diagram of the wire.
Figure-(1)
Concept used:
Expression of mass of the steel wire.
Here, density of the steel wire is
Expression of density of the steel wire.
Here, the specific weight of the steel wire is
Substitute
For section (1).
Substitute
Expression of volume of the steel wire after formation for section (1).
Here, the diameter of the formed wire is
For section (2).
Substitute
Expression of volume of the steel wire after formation for section (2).
For section (3).
Substitute
Expression of volume of the steel wire after formation for section (3).
For section (4).
Substitute
Expression of volume of the steel wire after formation for section (4).
Expression of Moment of inertia about
Here, the moment of inertia of section (1) about the
Expression of moment of inertia of section (1) about the
Expression of moment of inertia of section (2) about the
Expression of moment of inertia of section (3) about the
Expression of moment of inertia of section (4) about the
Expression of Moment of inertia about
Here, the moment of inertia of section (1) about the
Expression of moment of inertia of section (1) about the
Expression of moment of inertia of section (2) about the
Expression of moment of inertia of section (3) about the
Expression of moment of inertia of section (4) about the
Expression of Moment of inertia about
Here, the moment of inertia of section (1) about the
Expression of moment of inertia of section (1) about the
Expression of moment of inertia of section (2) about the
Expression of moment of inertia of section (3) about the
Expression of moment of inertia of section (4) about the
Mass products of inertia
Here. Mass products of inertia is
Mass products of inertia
Here. Mass products of inertia is
Mass products of inertia
Here. Mass products of inertia is
Calculation:
Substitute
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Substitute
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Substitute
Substitute
Thus, the mass moment of inertia of the wire with respect to
Substitute
Substitute
Substitute
Substitute
Substitute
Thus, the mass moment of inertia of the wire with respect to
Substitute
Substitute
Substitute
Substitute
Substitute
Product of inertia
Substitute
Substitute
Substitute
Mass moment of inertia of the machine component with respect to the axis through the origin by the unit vector is calculated as follows:
Conclusion:
The mass moment of inertia of the machine component with respect to the axis through the origin by the unit vector is
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Chapter B Solutions
Vector Mechanics For Engineers
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