Concept explainers
A 2-mm thick piece of sheet steel is cut and bent into the machine component shown. Knowing that the density of steel is 7850 kg/m3, determine the mass moment of inertia of the component with respect to each of the coordinate axes.
Fig. P9.137
Find the mass moment of inertia with respect to
Answer to Problem 9.137P
The mass moment of inertia with respect to
The mass moment of inertia with respect to
The mass moment of inertia with respect to
Explanation of Solution
Given information:
The thickness (t) of sheet steel is
The density
Calculation:
Divided the section into three geometric portions as shown below:
- Upper Flange
- Lower Flange
- Horizontal base
Find the mass of the upper flange component using the relation as shown below:
Here, V is volume of component of upper flange and A is the area of the section.
Find the area A of upper flange as shown below:
Here, b is the width of section and h is the height of section.
Substitute
Substitute
Find the mass of the lower flange component using the relation as shown below:
Here, V is volume of component of lower flange.
Find the area A of lower flange as shown below:
Here, b is the width of section and h is the height of section.
Substitute
Substitute
Find the moment of inertia about x axis of upper lower flange section as shown below:
Substitute
Find the moment of inertia about y axis of upper lower flange section as shown below:
Substitute
Find the moment of inertia about z axis of upper lower flange section as shown below:
Substitute
Find the mass of the horizontal base using the relation as shown below:
Find the area A of horizontal base as shown below:
Here, b is the width of section and h is the height of section.
Substitute
Substitute
Find the moment of inertia about x axis of horizontal base as shown in below:
Substitute
Find the moment of inertia about y axis of horizontal base as shown in below:
Substitute
Find the moment of inertia about z axis of horizontal base as shown in below:
Substitute
Find the total moment of inertia
Here,
Substitute
Thus, the mass moment of inertia with respect to
Find the total moment of inertia
Here,
Substitute
Thus, the mass moment of inertia with respect to
Find the total moment of inertia
Here,
Substitute
Thus, the mass moment of inertia with respect to
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