Name: area moment of inertia shift rule
Uploaded: 2025-05-22T09:06:05.000Z
Duration: 1 min 12 s
Channel: JoVE

What is the moment of inertia for the area of a thin plate about an arbitrary axis?
Consider a differential element at a certain distance from the centroidal axis parallel to the given axis. Its differential moment of inertia can be calculated.
Integrating the expression over the entire area yields the plate&#39;s moment of inertia about the planar axis.
Here, the first term represents the moment of inertia about the centroid, while the third term equals the total area. The second term is zero as this axis passes through the centroid.
The resulting expression gives the moment of inertia about the planar axis. Similarly, the moment of inertia about the other planar axis is obtained.
Summing these moments of inertia gives the polar moment of inertia.
So, the moment of inertia about any axis equals the moment of inertia about a centroidal axis parallel to it plus the product of the area and the square of the distance between the axes. The obtained expression is called the parallel axis theorem for an area.

parallel-axis theorem for an area

Parallel-Axis Theorem for an Area

The moment of inertia is a fundamental concept in mechanical engineering that plays a significant role in designing rotationally symmetric objects such as flywheels, gears, and other mechanical systems. In this context, we will discuss the moment of inertia of a flywheel rotating about its centroidal axis and how it relates to the moment of inertia about an axis parallel to it.
For a flywheel approximated as a solid disc, consider an infinitesimal differential element with an arbitrary distance from the centroidal axis. The moment of inertia for this differential element is calculated by multiplying the area of the differential element with the square of the distance from its centroidal axis. This equation is integrated over the entire disc area to determine the disc's moment of inertia.
<img alt="Static equilibrium formula \(I_x = \int_A y'^2 dA' + 2d_y \int_A y' dA' + d_y^2 \int_A dA'\)." src="/files/ftp_upload/14344/14344_PT_Equation_1.svg" style="text-align: center; height: 33px;" data-altupdatedat="1754904545000" data-alt="Moment of inertia equation, \(I_x\), integral formula, mathematical concept, educational use.">
Here, the first integral term represents the moment of inertia around the object's centroid, while the third integral term equals the total area. The second integral term is zero, as this axis passes through the centroid. The moment of inertia about an axis parallel to the centroidal axis equals the moment of inertia around the centroid plus the product of the area and distance squared. This is called the parallel-axis theorem for an area.
This theorem is an essential tool in mechanical engineering applications that involve the moment of inertia of an area. For example, when designing a flywheel or other rotating components, engineers must consider the moment of inertia around its centroidal axis and other axes parallel to the centroidal axis. This helps ensure that the flywheel's design is optimized for its intended application.
In addition, summing the moments of inertia along the planar axes gives the polar moment of inertia about an arbitrary point. This parameter is vital in mechanical engineering applications involving torques in different directions. By considering both the moment of inertia about the centroidal axis and the moment of inertia about the axes parallel to it, mechanical engineers can design more efficient and effective systems.

The moment of inertia is a fundamental concept in mechanical engineering that plays a significant role in designing rotationally symmetric objects such as ...

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Moment of Inertia

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area moment of inertia shift rule

The area moment of inertia changes when its axis moves away from the centroid. This concept is important in mechanical engineering for rotationally symmetric parts such as flywheels, gears, and similar systems. It helps describe how an area resists rotation about different axes.

A flywheel can be treated as a solid disc in this analysis. For a small area element, its contribution depends on the element's area and its distance from the centroidal axis. The moment of inertia is found by integrating these contributions across the full disc area.

The resulting expression includes three parts. The first integral gives the moment of inertia about the centroid. The second integral is zero because the axis passes through the centroid. The third integral equals the total area.

From this result, the moment of inertia about any axis parallel to the centroidal axis equals the centroidal moment plus the area times the square of the distance between the two axes. This is the parallel-axis theorem for an area. Engineers use it when analyzing a flywheel or other rotating component about axes that do not pass through the centroid.

The same idea also helps when finding the polar moment of inertia about an arbitrary point. Adding the moments of inertia along the planar axes gives this value. It is useful in mechanical engineering problems that involve torques acting in different directions. By comparing centroidal and parallel axes, engineers can design more efficient systems.

Area Moment of Inertia Shift Rule

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...

The area moment of inertia changes when its axis moves away from the centroid. This concept is important in mechanical engineering for rotationally symmetric parts such as flywheels, gears, and similar systems. It helps describe how an area resists rotation about different axes. A flywheel can be treated as a solid disc in this analysis. For a small area element, its contribution depends on the element's area and its distance from the centroidal axis. The moment of inertia is found by...

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How Area Shape Affects Beam Stiffness

The second moment of area helps explain how a shape resists bending. It is also called the moment of inertia of area, and it is a key measure of stiffness. Engineers use it to understand how an object may deform under load. To estimate the second moment of area along a chosen axis, the area is concentrated into a thin strip. That strip is placed parallel to the axis being studied. The distance from the strip to that axis is found using the radius of gyration. The radius of gyration is the...

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Finding Principal Axes in Area Inertia

Moments of inertia of area and the product of inertia are used in mechanics to describe how an area responds about different axes. They help with the stability and performance of structures and components. To study an inclined axis, coordinate transformation relations are used to calculate both the moment of inertia and the product of inertia for the area. The principal axes are the special axes that give the maximum and minimum values of the moment of inertia. Their orientation is found by...

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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...

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Composite Area Centroids and Bending Stiffness

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

moments-of-inertia-for-an-area-about-inclined-axes

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

The mass moment of inertia describes how much a rigid body resists angular acceleration. It applies to rotating systems such as fans, wind turbines, and many machines used in industry. Engineers use it to judge how easily an object can start or change its spin. For an arbitrary object, a small mass element contributes to the total moment of inertia based on its mass and its distance from the rotation axis. The farther the mass lies from that axis, the larger its contribution becomes. Adding...

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 the area moment of inertia changes with axis location and how the parallel-axis theorem applies to flywheels, discs, and other rotating parts.

Area Moment of Inertia Shift Rule | JoVE K12

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Area Moment of Inertia Shift Rule

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