10.5:
Moments of Inertia: Problem Solving
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The second moment of an area, also known as the moment of inertia of an area, is a geometric property of a shape that reflects its resistance to change. The moment of inertia of an area can be calculated for both two-dimensional and three-dimensional shapes. The moment of inertia of an area is calculated by taking the sum of the product of the area and the square of its distance from a chosen axis of rotation. For two-dimensional shapes, the moment of inertia can be expressed as a single equation in terms of the shape's x and y coordinates.
To determine the moment of inertia for a schematic triangle along the centroidal axis, as shown above, we must begin by considering a differential strip at a certain distance from the base parallel to the centroidal axis. The differential moment of inertia equals the distance squared multiplied by the differential area.
The length of the strip is estimated using the law of similar triangles. The strip's length is proportional to its distance from the base.
Rewriting the differential area in terms of the length and width of the differential strip and using the expression for the strip length, the equation is modified. Now, integrating the differential moment of inertia along the entire height gives the total moment of inertia along the base.
The centroid of a triangle is located at one-third of the triangle's height from its base. Using the parallel axis theorem, the moment of inertia along the centroid equals the moment of inertia along the base minus the product of the triangle's area and the centroidal axis distance squared.
By substituting the relevant values, the moment of inertia of the triangle along the centroidal axis can be obtained.
Triangular plates are commonly used in the design of the tail section of aircraft. This section of the aircraft is responsible for providing stability and control during flight. Triangular plates are often used to form the vertical and horizontal stabilizers, which help keep the aircraft stable and pointing in the right direction.