Concept explainers
9.23 and 9.24 Determine the polar moment of inertia and the polar radius of gyration of the shaded area shown with respect to point P.
Fig. P9.24
Find the polar moment of inertia and polar radius of gyration of the shaded area with respect to point P.
Answer to Problem 9.24P
The polar moment of inertia of the shaded area with respect to point P is
The polar radius of gyration of the shaded area with respect to point P is
Explanation of Solution
Calculation:
Sketch the horizontal strip along circular portion as shown in Figure 1.
Write the curve equation of circle as follows:
Modify Equation (1).
Determine the area of the strip element
Substitute
Find the shaded area (A) using the relation:
Substitute
Consider
Differentiate both sides of the Equation.
Substitute
Determine the moment of inertia
Substitute
Integrate Equation (5) with respect to y.
Consider
Differentiate both sides of the Equation.
Substitute
Determine the moment of inertia
Substitute
Integrate Equation (8) with respect to y.
Consider
Differentiate both sides of the Equation.
Substitute
Find the polar moment of inertia
Here,
Substitute
Thus, the polar moment of inertia of the shaded area with respect to point P is
Find the polar radius of gyration
Here,
Substitute
Thus, the polar radius of gyration of the shaded area with respect to point P is
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Chapter 9 Solutions
Vector Mechanics for Engineers: Statics and Dynamics
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