Name: moment and product of inertia basics
Uploaded: 2024-03-07T13:22:27.000Z
Duration: 1 min 23 s
Channel: JoVE

The moment of inertia for a differential element of a rigid body can be calculated by multiplying the mass of the element by the square of the shortest distance from any one of the three-coordinate axes to the element.
By integrating this expression over the entire mass of the body, the moment of inertia of the body can be determined.
Similarly, the moment of inertia about the other two axes can be found. The moment of inertia is always a positive quantity.
Additionally, the product of inertia for a differential element in relation to a pair of perpendicular planes is defined as the product of the mass of the element and the perpendicular distance from the plane to the element.
Integrating this over the entire mass, the product of the inertia of the body is calculated. The product of inertia can be positive, negative, or zero.
If the mass is symmetric about one or both orthogonal planes, then the product of inertia about such planes is always zero.

moments and product of inertia

Moments and Product of Inertia

The calculation of the moment of inertia for a differential element within a rigid body involves multiplying the element's mass by the square of the minimum distance from any one of the three-coordinate axes to the said element. This is a process that can be extended to cover the entire mass of the body by simply integrating the expression, thereby ascertaining the body's moment of inertia.
<img alt="Moment of inertia formula, Ix=&#8747;mr&#178;x, mathematical derivation, physics equation." src="/files/ftp_upload/15545/15545_PT_Equation_001.svg" style="text-align: center; height: 42px;" data-altupdatedat="1762173717000" data-alt="Equation 1" />
The same process can be applied to determine the moment of inertia in relation to the other two axes. It is important to note that the moment of inertia is invariably a positive quantity.
Furthermore, there is also a product of inertia related to a differential element and a pair of perpendicular planes. This is defined as the multiplication of the element's mass by the perpendicular distance from the plane to the element. By integrating this across the body's entire mass, one can calculate the body's product of inertia.
<img alt="Static equilibrium formula, Ix,y=Iy,x integral representing product of coordinates, physics concept." src="/files/ftp_upload/15545/15545_PT_Equation_002.svg" style="text-align: center; height: 42px;" data-altupdatedat="1762173717000" data-alt="Equation 2" />
A similar analysis can be done for the remaining two planes. Unlike the moment of inertia, the product of inertia can either be positive, negative, or zero.
For bodies where the mass distribution is symmetric about one or both orthogonal planes, the product of inertia about such planes will always be zero. This symmetry plays a crucial role in determining the product of inertia. Overall, these calculations provide insights into the dynamic properties of a rigid body, underlining the importance of understanding the concepts of moment of inertia and product of inertia.

The moment of inertia for a differential element of a rigid body can be calculated by multiplying the mass of the element by the square of the shortest ...

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3-Dimensional Kinetics of a Rigid Body

KEY TERMS AND CONCEPTS

moment and product of inertia basics

Moment of inertia and product of inertia describe how mass is distributed in a rigid body. They help show how the body responds about coordinate axes and planes. For a small mass element, the moment of inertia is found by multiplying the element's mass by the square of the shortest distance to one of the three coordinate axes.

That same idea is extended to the whole body by integration. Adding the contributions of all the small mass elements gives the body's moment of inertia about that axis. The same process can be used for the other two axes. The moment of inertia is always a positive quantity.

The product of inertia is defined in a similar way for a differential element and a pair of perpendicular planes. It is found by multiplying the element's mass by the perpendicular distance from the plane to the element. When the expression is integrated over the entire body, the product of inertia is obtained for that set of planes.

The other two plane pairs can be treated the same way. Unlike the moment of inertia, the product of inertia can be positive, negative, or zero. When a body has mass distributed symmetrically about one or both orthogonal planes, the product of inertia about those planes is zero. This symmetry is important in understanding the dynamic behavior of a rigid body.

Moment and Product of Inertia Basics

Moment of inertia and product of inertia describe how mass is distributed in a rigid body. They help show how the body responds about coordinate axes and planes. For a small mass element, the moment of inertia is found by multiplying the element's mass by the square of the shortest distance to one of the three coordinate axes. That same idea is extended to the whole body by integration. Adding the contributions of all the small mass elements gives the body's moment of inertia about that axis.

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Principal Axes and Moments of Inertia

The inertia tensor describes how mass is spread out in a rigid body and how that body resists rotation. It is written as a 3×3 matrix, and each entry represents a moment of inertia about a particular axis. This makes the tensor a compact way to study rotational behavior in three dimensions. The diagonal entries show the moments of inertia about the principal axes. These principal axes are the directions where the object has the least resistance to rotation. A smaller moment of inertia along...

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Calculating Moment of Inertia on Any Axis

Moment of inertia can be calculated about any chosen axis, not just a principal axis. It depends on the mass distribution of the object along that axis. This makes it useful when an object rotates in a real machine or other setup where balance and stability matter. To find the moment of inertia about an arbitrary axis, the perpendicular distance from the axis to each mass element is needed. That distance is obtained with a cross product between the unit vector that gives the axis direction and...

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Angular momentum relative to point P is found by adding the motion of every mass element in a rigid body. The body has mass m and a center of mass at point G. It rotates in an inertial reference frame, and the angular momentum at P comes from the cross product of each position vector and its linear momentum. To build the full expression, the velocity of each mass element is written as two parts. One part is the translational velocity of the body. The other part is the relative velocity caused...

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Angular Momentum in Principal Axes

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Impulse and Angular Momentum in Rigid Bodies

Impulse and angular momentum help describe a rigid body moving in a plane. This motion combines translation and rotation. The topic uses Newton's second law to connect force, momentum, and motion. For translational motion, the equation is written for the center of mass, labeled G. When this equation is multiplied by a small time interval, dt, and integrated, it gives the principle of linear impulse. This principle shows that a change in momentum is proportional to the impulse applied to the...

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Rigid Body Energy from Center and Rotation

Rigid body kinetic energy in planar motion can be written using the body’s center of mass and its rotation. The solid object moves in general planar motion, and its center of mass is marked at point G. To find the kinetic energy of the i-th particle, the relative velocity definition is used with the position vector rA from point A to that particle. The kinetic energy of the whole body is built from the particles. First, a scalar product is taken. Then the expression is written in integral form.

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Rigid Body Rotation in Euler’s Frame

Euler’s equations of motion describe how a rigid body rotates when measured in a moving frame attached to the body. The body rotates with angular velocity ω in an inertial frame, and the attached frame rotates with angular velocity Ω. Together, these two frames help describe the motion of the rigid body around its center of mass. The total moment about the center of mass is found by adding two parts. One part is the rate of change of angular momentum about the center of mass in the rotating...

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Torque-free motion describes the movement of a rigid body when no external torques act on it. It can be seen in places with no outside forces or friction, such as outer space. Mars is one example of this kind of motion because it rotates in space without external torque. Mars is an axisymmetric object, which means it has an axis of symmetry. That symmetry defines the rotating z-axis. The rotating frame is set so that Mars&amp;#39;s center of mass is at the origin. This setup makes the moments...

key terms and concepts

Learn how moment of inertia and product of inertia are calculated from small mass elements, why symmetry matters, and how these values describe rigid body motion.

Moment and Product of Inertia Basics | JoVE K12

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

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Moment and Product of Inertia Basics

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