Name: calculating moment of inertia on any axis
Uploaded: 2025-05-22T09:06:17.000Z
Duration: 1 min 20 s
Channel: JoVE

The moment of inertia is commonly discussed in relation to principal axes, but it can also be calculated for any arbitrary axis.
When considering an arbitrary axis, the moment of inertia is determined by integrating the mass distribution of the object along that axis.
Here, the perpendicular distance between an arbitrarily chosen axis and the considered mass distribution is given by the cross-product of the unit vector that defines the direction of the axis and the position vector for the mass element.
Performing the dot product operation and expanding the brackets gives the expression for the moment of inertia.
Using the definitions of the moment of inertia and the product of inertia along the different axes, the moment of inertia along an arbitrary axis can be generalized.
If the inertia tensor is defined with respect to the xyz axes, then the moment of inertia about an arbitrary axis can be calculated if the direction cosines of the axis are known.

moment of inertia about an arbitrary axis

Moment of Inertia about an Arbitrary Axis

The moment of inertia is typically associated with principal axes, but it can also be computed for any random axis. When an arbitrary axis is under consideration, the moment of inertia is determined by integrating the mass distribution of the object along that specific axis. It is crucial in applications like the design of machinery, where components rotate about various axes, and balance and stability are essential.
In this scenario, the perpendicular distance between the chosen arbitrary axis and the mass distribution being considered is derived from the cross-product of the unit vector, which defines the direction of the axis and the position vector for the mass element.
When a dot product operation is performed, and the brackets are expanded, it results in the expression for the moment of inertia. Utilizing the definitions of the moment of inertia and the product of inertia along various axes, the moment of inertia along an arbitrary axis can be generalized.
Suppose the inertia tensor is defined with respect to the XYZ axes. In that case, calculating the moment of inertia about an arbitrary axis becomes possible if the direction cosines of the axis are known. This approach allows for a more comprehensive understanding of the moment of inertia, beyond the limitations of considering only principal axes.

The moment of inertia is typically associated with principal axes, but it can also be computed for any random axis. When an arbitrary axis is under ...

moment-of-inertia-about-an-arbitrary-axis

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Mechanical Engineering

Unit 1

3-Dimensional Kinetics of a Rigid Body

KEY TERMS AND CONCEPTS

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 the position vector of the mass element. The result is then used in the integration for moment of inertia.

When the dot product is expanded, it leads to the expression for moment of inertia about the chosen axis. The same idea can be written in a general form using the moment of inertia and product of inertia along different axes. If the inertia tensor is known with respect to the XYZ axes, the moment of inertia about any arbitrary axis can be found from the direction cosines of that axis. This gives a broader way to study rotation beyond only the principal axes.

Calculating Moment of Inertia on Any Axis

calculating-moment-of-inertia-on-any-axis

moments and product of inertia

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.

inertia tensor

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...

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...

angular momentum about an arbitrary axis

Angular Momentum Relative to Point P

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...

angular momentum and principle axes of inertia

angular-momentum-and-principle-axes-of-inertia

Angular Momentum in Principal Axes

Angular momentum in a rigid body can be written using rectangular coordinates. The body’s angular momentum comes from the cross-product of each mass element’s position vector with the cross-product of the body’s angular velocity and that position vector. This gives a compact way to describe how a solid structure moves in rotation. To work with the equation more easily, the XYZ axes can be chosen as a new set of rectangular axes. These axes may be inclined at any angle to the reference frame.

principle of impulse and moment

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...

kinetic energy for a rigid body

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.

equation of motion for a rigid body

Rigid Body Motion: Force and Moment Balance

Rigid body motion is described with both translational motion and rotational motion about the center of mass, point G. The center of mass is the point where Newton's Second Law applies to translation. It gives the linear motion of the body. The same body also has rotational motion about point G. The combined moments about the center of mass equal the rate of change of angular momentum. A moment is the turning effect caused by a force, and an external force applied away from the center of mass...

euler equations of motion

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...

torque free motion

Mars Spin Under No Torque

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 is calculated for an arbitrary axis using cross products, dot products, and the inertia tensor with direction cosines.

Calculating Moment of Inertia on Any Axis | JoVE K12

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Calculating Moment of Inertia on Any Axis

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