Name: moment of inertia for triangles
Uploaded: 2023-09-22T11:49:34.000Z
Duration: 1 min 14 s
Channel: JoVE

Consider airplanes, where the wing components are triangular plates. Approximating this component as a schematic triangle, what is the moment of inertia about the centroidal axis?
The centroid of a triangle is located at one-third of the triangle&#39;s height from its base.
Consider a differential strip at a certain distance from the base parallel to the centroidal axis. This strip&#39;s differential moment of inertia equals the square of the distance from the axis times the differential area.
The differential area is the product of the strip&#39;s length and width. The strip&#39;s length is estimated considering the law of similar triangles.
Integrating the differential moment of inertia along the entire height gives the moment of inertia about the base.
Using parallel axis theorem, the moment of inertia about the centroid equals the moment of inertia about the base minus the product of the traingle&#39;s area and the square of the distance from the centroidal axis.
Substituting the values, the moment of inertia of the triangle about the centroidal axis is obtained.

moments of inertia: problem solving

Moments of Inertia: Problem Solving

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.
<img alt="Triangle centroid diagram, axes labeled x, y, distances b, h, l, centroid C coordinates xy'." src="/files/ftp_upload/14346/14346_PT_Fig_1.svg" style="text-align: center; height: 250px;" data-altupdatedat="1761830439000" data-alt="Figure 1">
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.
<img alt="static equilibrium equation dIx=y²dA' diagram; mechanical analysis, educational use" src="/files/ftp_upload/14346/14346_PT_Equation_1.svg" style="text-align: center; height: 16px;" data-altupdatedat="1761830439000" data-alt="Equation 1">
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.
<img alt="Statics formula l/b=(h-y)/h, equilibrium concept, mathematical equation." src="/files/ftp_upload/14346/14346_PT_Equation_2.svg" style="text-align: center; height: 30px;" data-altupdatedat="1761830439000" data-alt="Equation 2">
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.
<img alt="Static equilibrium formula, Ix integration equation diagram, mathematical principles calculation." src="/files/ftp_upload/14346/14346_PT_Equation_3.svg" style="text-align: center; height: 33px;" data-altupdatedat="1761830439000" data-alt="Equation 3">
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.
<img alt="Parallel axis theorem, Ix' = Ix - A(h/3)^2, formula diagram for moment of inertia calculations." src="/files/ftp_upload/14346/14346_PT_Equation_4.svg" style="text-align: center; height: 33px;" data-altupdatedat="1761830439000" data-alt="Equation 4">
By substituting the relevant values, the moment of inertia of the triangle along the centroidal axis can be obtained.
<img alt="Cross-section bending equation: I̅x′ = (bh³)/36; structural mechanics diagram." src="/files/ftp_upload/14346/14346_PT_Equation_5.svg" style="text-align: center; height: 33px;" data-altupdatedat="1761830439000" data-alt="Equation 5">
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.

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. ...

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Moment of Inertia

KEY TERMS AND CONCEPTS

moment of inertia for triangles

The moment of inertia of an area describes how a shape resists change. It is also called the second moment of area. This property can be calculated for both two-dimensional and three-dimensional shapes.

For a shape in two dimensions, the moment of inertia is found by adding the area multiplied by the square of its distance from a chosen axis of rotation. In this case, the equation can be written in terms of the shape's x and y coordinates. The setup shows how the shape's geometry affects the result.

To find the moment of inertia of a schematic triangle along its centroidal axis, the calculation starts with a thin differential strip. The strip is placed at a distance from the base and runs parallel to the centroidal axis. Its differential moment of inertia equals the square of the distance times the differential area.

The strip length is estimated with the law of similar triangles. This makes the strip length proportional to its distance from the base. The differential area is then rewritten using the strip's length and width. After that, the expression is integrated across the full height to find the total moment of inertia along the base.

The centroid of a triangle lies one-third of the triangle's height from the base. Using the parallel axis theorem, the moment of inertia about the centroid is found from the moment of inertia about the base minus the triangle's area times the square of the distance between the axes. After substituting the values, the moment of inertia along the centroidal axis is obtained.

Triangular plates are often used in aircraft tail sections. That part of the aircraft provides stability and control during flight. These plates help form the vertical and horizontal stabilizers, which keep the aircraft stable and pointed in the right direction.

Moment of Inertia for Triangles

moments of inertia for areas

Area Moments in Shape Design

Area moments measure how a shape resists change when it is rotated or loaded. The second moment of area, also called the moment of inertia of an area, is a geometric property of a shape. It is written as a single number and can be found for both two-dimensional and three-dimensional shapes. To calculate it, add the area of each part of the shape multiplied by the square of its distance from a chosen axis of rotation. This makes the result depend on how far the area lies from that axis. The...

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Finding Principal Axes in Area Inertia

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The moment of inertia of an area describes how a shape resists change. It is also called the second moment of area. This property can be calculated for both two-dimensional and three-dimensional shapes. For a shape in two dimensions, the moment of inertia is found by adding the area multiplied by the square of its distance from a chosen axis of rotation. In this case, the equation can be written in terms of the shape's x and y coordinates. The setup shows how the shape's geometry affects the...

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Composite areas are structures made from several basic shapes joined together. These shapes often include rectangles, triangles, circles, and other simple forms. Their combined shape acts as a single structure. Finding the second moment of area is important when studying how stiff a structure is. The second moment of area, also called the moment of inertia, measures resistance to bending. It depends on how the area is spread around a reference axis. To find the second moment of area for a...

product of inertia for an area

Centroidal Product of Inertia in Area Analysis

Mechanical engineering uses the product of inertia for an area to help judge how a shape responds to force. This calculation describes how the mass of a structure is spread around its centroid, or center point. It also helps show how the structure may resist rotational forces and how those forces can change the magnitude and direction of stress. To find the product of inertia for an arbitrary area, the area is first divided into small two-dimensional elements. A Cartesian coordinate system is...

moments of inertia for an area about inclined axes

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Inclined-Axis Moment and Product of Inertia

Inclined-axis moment and product of inertia are important in physics and engineering. They help describe how an area is distributed around axes that are not aligned with the original coordinate system. For an irregular area, these values support better design and analysis. The moment of inertia can be found by integrating the contribution from a very small area element. When a second coordinate system is tilted by an angle, transformation relations connect the inclined axes to the original...

mohr&#39;s circle for moments of inertia

Finding Principal Moments with Mohr’s Circle

Mohr’s circle is a graphical method for finding an area’s principal moments of inertia. It uses a rectangular coordinate system to plot the moments of inertia and the product of inertia. This makes the key values easier to read from the graph. The center of Mohr’s circle comes from averaging the moments of inertia about the x- and y-axis. The radius depends on the moments and products of inertia. Where the circle crosses the horizontal axis, the principal moments of inertia are found. At those...

mohr&#39;s circle for moments of inertia: problem solving

mohr-s-circle-for-moments-of-inertia-problem-solving

Finding Principal Axes with Mohr’s Circle

Mohr's circle is a graphical method for finding an area's principal moments of inertia and the orientation of its principal axes. It works by plotting the moments of inertia and the product of inertia on a rectangular coordinate system. For a rectangular beam, the moments of inertia about the x- and y-axes are 2.5(10^7) mm^4 and 7.5(10^7) mm^4, and the product of inertia is 1.5(10^7) mm^4. The first step is to locate the center of the circle. It is found by taking the average of the x- and...

mass moment of inertia

Moment of Inertia in a Solid Sphere

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mass moment of inertia: problem solving

Calculating Cylinder Moment of Inertia

Calculating the moment of inertia of a cylinder is important in engineering and automotive design. It helps show how changes in mass, radius, and shape can affect performance. For a wheel axle, a solid cylinder model can be used with both longitudinal and perpendicular axes. One step in the calculation uses a thin disc drawn parallel to the circular face of the cylinder. The disc has a moment of inertia equal to one-fourth the product of mass and radius squared. The disc is then shifted away...

key terms and concepts

Learn how to calculate the moment of inertia for a triangle using differential strips, similar triangles, and the parallel axis theorem, with an aircraft example.

Moment of Inertia for Triangles | JoVE K12

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Moment of Inertia for Triangles

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