Consider the cross-section below. A compressive load P is applied at any point (ex, ey) with respect to the principal axes x and y. The moment of P about these axes are respectively

$M_x = Pe_y$     and     $M_y = Pe_x$

Area, moment of inertia, and radius of gyration of parabolic section

Situation
Given the parabola 3x2 + 40y – 4800 = 0.

Part 1: What is the area bounded by the parabola and the X-axis?
A. 6 200 unit2
B. 8 300 unit2
C. 5 600 unit2
D. 6 400 unit2

Part 2: What is the moment of inertia, about the X-axis, of the area bounded by the parabola and the X-axis?
A. 15 045 000 unit4
B. 18 362 000 unit4
C. 11 100 000 unit4
D. 21 065 000 unit4

Part 3: What is the radius of gyration, about the X-axis, of the area bounded by the parabola and the X-axis?
A. 57.4 units
B. 63.5 units
C. 47.5 units
D. 75.6 units

818 Hollow square section | Moment of Inertia and Radius of Gyration

Problem 818
A hollow square cross section consists of an 8 in. by 8 in. square from which is subtracted a concentrically placed square 4 in. by 4 in. Find the polar moment of inertia and the polar radius of gyration with respect to a z axis passing through one of the outside corners.

817 Hollow Tube | Moment of Inertia and Radius of Gyration

Problem 817
Determine the moment of inertia and radius of gyration with respect to a polar centroidal axis of the cross section of a hollow tube whose outside diameter is 6 in. and inside diameter is 4 in.

816 Polar moment of inertia and radius of gyration at one corner of rectangle

Problem 816
A rectangle is 3 in. by 6 in. Determine the polar moment of inertia and the radius of gyration with respect to a polar axis through one corner.

Moment of Inertia and Radius of Gyration

Moment of Inertia
Moment of inertia, also called the second moment of area, is the product of area and the square of its moment arm about a reference axis.

Moment of inertia about the x-axis:

$\displaystyle I_x = \int y^2 \, dA$