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. P9.23
Find the polar moment of inertia and polar radius of gyration of the shaded area shown with respect to point P.
Answer to Problem 9.23P
The polar moment of inertia of the shaded area shown with respect to point P is
The polar radius of gyration of the shaded area shown with respect to point P is
Explanation of Solution
Given information:
The equation of the curve is
The equation of the curve is
Calculation:
Sketch the vertical strip shaded portion as shown in Figure 1.
Write the curve
Write the curve
Refer to Figure 1.
At
For
Find the constant
Substitute
Substitute 0 for x and a for y in Equation (2).
Refer to Figure 1.
At
For
Find the constant
Substitute
Substitute
Substitute
Determine the area of the strip element
Substitute
Find the shaded area (A) using the relation:
Substitute
Determine the moment of inertia
Refer Equation 9.2 in section 9.1B determining the moment of inertia of an area by integration:
Substitute
Integrate Equation (6) with respect to x.
Determine the moment of inertia
Integrate Equation (7) with respect to x.
Find the polar moment of inertia
Here,
Substitute
Thus, the polar moment of inertia of the shaded area shown with respect to point P is
Find the polar radius of gyration of the shaded area shown with respect to point P.
Here,
Substitute
Thus, the polar radius of gyration of the shaded area shown with respect to point P is
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Chapter 9 Solutions
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