Moment of inertia

The moment of inertia, otherwise known as the angular mass or rotational inertia, of a rigid body is a tensor that determines the torque needed for a desired angular acceleration about a rotational axis; similar to how mass determines the force needed for a desired acceleration.

Tensors are important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (stress, elasticity, fluid mechanics, moment of inertia, ...), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), or general relativity (stress–energy tensor, curvature tensor, ... ) and others.

The moment of inertia, otherwise known as the angular mass or rotational inertia, of a rigid body is a tensor that determines the torque needed for a desired angular acceleration about a rotational axis; similar to how mass determines the force needed for a desired acceleration. When a body is free to rotate around an axis, torque must be applied to change its angular momentum.

The definition of torque states that one or both of the angular velocity or the moment of inertia of an object are changing.

Its simplest definition is the second moment of mass with respect to distance from an axis.

If the function represents physical density, then the zeroth moment is the total mass, the first moment divided by the total mass is the center of mass, and the second moment is the rotational inertia.

When a body is free to rotate around an axis, torque must be applied to change its angular momentum. Moment of inertia also appears in momentum, kinetic energy, and in Newton's laws of motion for a rigid body as a physical parameter that combines its shape and mass.

For continuous rigid bodies, though, the spin angular velocity ω is proportional but not always parallel to the angular momentum of the object, making the constant of proportionality I (called the moment of inertia) a second-rank tensor rather than a scalar.

Moment of inertia also appears in momentum, kinetic energy, and in Newton's laws of motion for a rigid body as a physical parameter that combines its shape and mass.

where M is the total mass and I is the moment of inertia about an axis perpendicular to the movement of the rigid system and through the center of mass.

The moment of inertia, otherwise known as the angular mass or rotational inertia, of a rigid body is a tensor that determines the torque needed for a desired angular acceleration about a rotational axis; similar to how mass determines the force needed for a desired acceleration.

The angular momentum with respect to the center of mass is the same as without translation: at any time it is equal to the inertia tensor times the angular velocity. When the angular velocity is expressed with respect to a coordinate system coinciding with the principal axes of the body, each component of the angular momentum is a product of a moment of inertia (a principal value of the inertia tensor) times the corresponding component of the angular velocity; the torque is the inertia tensor times the angular acceleration.

The moment of inertia, otherwise known as the angular mass or rotational inertia, of a rigid body is a tensor that determines the torque needed for a desired angular acceleration about a rotational axis; similar to how mass determines the force needed for a desired acceleration.

where m is the mass of the particle and I is its moment of inertia.

In 1673 Christiaan Huygens introduced this parameter in his study of the oscillation of a body hanging from a pivot, known as a compound pendulum.

where I\; is the moment of inertia of the pendulum about the pivot point,

Its simplest definition is the second moment of mass with respect to distance from an axis.

The ratio of torque and angular acceleration (how difficult it is to start, stop, or otherwise change rotation) is given by the moment of inertia: T = I\alpha.

The moment of inertia, otherwise known as the angular mass or rotational inertia, of a rigid body is a tensor that determines the torque needed for a desired angular acceleration about a rotational axis; similar to how mass determines the force needed for a desired acceleration.

As a consequence of Newton's First Law of Motion, there exists rotational inertia that ensures that all bodies maintain their angular momentum unless acted upon by an unbalanced torque.

The moment of inertia of the body about its center of mass, I_C, is then calculated using the parallel axis theorem to be

The parallel axis theorem, also known as Huygens–Steiner theorem, or just as Steiner's theorem, named after Christiaan Huygens and Jakob Steiner, can be used to determine the mass moment of inertia or the second moment of area of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's center of gravity and the perpendicular distance between the axes.

The term moment of inertia was introduced by Leonhard Euler in his book Theoria motus corporum solidorum seu rigidorum in 1765, and it is incorporated into Euler's second law.

is the moment of inertia of the body about its center of mass.See also Euler's equations (rigid body dynamics).

A list of moments of inertia formulas for standard body shapes provides a way to obtain the moment of inertia of a complex body as an assembly of simpler shaped bodies.

In physics and applied mathematics, the mass moment of inertia, usually denoted by, measures the extent to which an object resists rotational acceleration about a particular axis, and is the rotational analogue to mass. Mass moments of inertia have units of dimension ML 2 ([mass] × [length] 2 ).

In 1673 Christiaan Huygens introduced this parameter in his study of the oscillation of a body hanging from a pivot, known as a compound pendulum.

By his study of the oscillation period of compound pendulums Huygens made pivotal contributions to the development of the concept of moment of inertia.

where k is known as the radius of gyration.

The radius of gyration about a given axis can be computed in terms of the mass moment of inertia around that axis, and the total mass m;

Moment of inertia also appears in momentum, kinetic energy, and in Newton's laws of motion for a rigid body as a physical parameter that combines its shape and mass.

I\, is the body's moment of inertia, equal to.

Specifically, it is the second moment of mass with respect to the orthogonal distance from an axis (or pole).

The moment of inertia is the 2nd moment of mass: I = r^2 m for a point mass, for a collection of point masses, or for an object with mass distribution . Note that the center of mass is often (but not always) taken as the reference point.

Flywheels resist changes in rotational speed by their moment of inertia.

A simple pendulum that has the same natural frequency as a compound pendulum defines the length L from the pivot to a point called the center of oscillation of the compound pendulum.

where I is the moment of inertia around the CM.

The moment of inertia matrix in body-frame coordinates is a quadratic form that defines a surface in the body called Poinsot's ellipsoid.

If the rigid rotor is symmetric (has two equal moments of inertia), the vector describes a cone (and its endpoint a circle).

Another expression replaces the summation with an integral,

In mechanics, the moment of inertia is calculated as the volume integral (triple integral) of the density weighed with the square of the distance from the axis:

Rotational energy

I \ is moment of inertia around the axis of rotation.

Note on second moment of area: The moment of inertia of a body moving in a plane and the second moment of area of a beam's cross-section are often confused.

Moment of inertia

defines an ellipsoid in the body frame.

The moments of inertia of an ellipsoid of uniform density are:

The moment of inertia, otherwise known as the angular mass or rotational inertia, of a rigid body is a tensor that determines the torque needed for a desired angular acceleration about a rotational axis; similar to how mass determines the force needed for a desired acceleration.