Concept explainers
Determine for the quarter ellipse of Prob. 9.67 the moments of inertia and the product of inertia with respect to new axes obtained by rotating the x and y axes about O (a) through 45° counterclockwise, (b) through 30° clockwise.
9.67 through 9.70 Determine by direct integration the product of inertia of the given area with respect to the x and y axes.
Fig. P9.67
(a)
Find the moment of inertia and product of inertia for quarter ellipse with respect new centroid axes obtained by rotating the x and y axes about O through
Answer to Problem 9.79P
The moment of inertia for quarter ellipse with respect to new centroid axes obtained by rotating the x about O through
The moment of inertia for quarter ellipse with respect to new centroid axes obtained by rotating the y about O through
The product of inertia for quarter ellipse with respect to new centroid axes obtained by rotating the x and y about O through
Explanation of Solution
Calculation:
Sketch the quarter ellipse as shown in Figure 1.
Refer to Figure 9.12 “Moments of inertia of common geometric Shapes” in the textbook.
Find the moment of inertia
Here, a is moments and products of area for a quarter of a circle of radius.
Substitute
Find the moment of inertia
Substitute
Refer to problem 9.67.
Write the curve Equation as shown below:
Modify Equation (3).
Select a vertical strip as differential element of area.
Applying the parallel axis theorem.
Here,
Using the property of symmetry about x and y axis.
Express the variables in terms of x and y.
Find the coordinate of centroid element
Substitute
Consider the element strip as follows:
Integrating
Find the value of
Find the value of
Find the moment of inertia for quarter ellipse with respect new centroid axes obtained by rotating the x about O through
Refer to Equation 9.18 in section 9.3B in the textbook.
Substitute
Thus, the moment of inertia for quarter ellipse with respect new centroid axes obtained by rotating the x about O through
Find the moment of inertia for quarter ellipse with respect new centroid axes obtained by rotating the y about O through
Refer to Equation 9.19 in section 9.3B in the textbook.
Substitute
Thus, the moment of inertia for quarter ellipse with respect new centroid axes obtained by rotating the y about O through
Find the product of inertia for quarter ellipse with respect new centroid axes obtained by rotating the x and y about O through
Substitute
Thus, the product of inertia for quarter ellipse with respect new centroid axes obtained by rotating the x and y about O through
(b)
Find the moment of inertia and product of inertia with respect new centroid axes obtained by rotating the x and y axes about O through
Answer to Problem 9.79P
The moment of inertia for quarter ellipse with respect new centroid axes obtained by rotating the x about O through
The moment of inertia for quarter ellipse with respect new centroid axes obtained by rotating the y about O through
The product of inertia for quarter ellipse with respect new centroid axes obtained by rotating the x and y about O through
Explanation of Solution
Calculation:
Find the moment of inertia for quarter ellipse with respect new centroid axes obtained by rotating the x about O through
Refer to Equation 9.18 in section 9.3B in the textbook.
Substitute
Thus, the moment of inertia for quarter ellipse with respect new centroid axes obtained by rotating the x about O through
Find the moment of inertia for quarter ellipse with respect new centroid axes obtained by rotating the y about O through
Refer to Equation 9.19 in section 9.3B in the textbook.
Substitute
Thus, the moment of inertia for quarter ellipse with respect new centroid axes obtained by rotating the y about O through
Find the product of inertia for quarter ellipse with respect new centroid axes obtained by rotating the x and y about O through
Substitute
Thus, the product of inertia for quarter ellipse with respect new centroid axes obtained by rotating the x and y about O through
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