Moment of inertia

Flywheels have large moments of inertia to smooth out rotational motion.
Tightrope walkers use the moment of inertia of a long rod for balance as they walk the rope. Samuel Dixon crossing the Niagara River in 1890.
Spinning figure skaters can reduce their moment of inertia by pulling in their arms, allowing them to spin faster due to conservation of angular momentum.
Pendulums used in Mendenhall gravimeter apparatus, from 1897 scientific journal. The portable gravimeter developed in 1890 by Thomas C. Mendenhall provided the most accurate relative measurements of the local gravitational field of the Earth.
The cylinders with higher moment of inertia roll down a slope with a smaller acceleration, as more of their potential energy needs to be converted into the rotational kinetic energy.
This 1906 rotary shear uses the moment of inertia of two flywheels to store kinetic energy which when released is used to cut metal stock (International Library of Technology, 1906).
A 1920s John Deere tractor with the spoked flywheel on the engine. The large moment of inertia of the flywheel smooths the operation of the tractor.

Quantity that determines the torque needed for a desired angular acceleration about a rotational axis, akin to how mass determines the force needed for a desired acceleration.

- Moment of inertia

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Flywheel

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A Landini tractor with exposed flywheel
A mass-produced flywheel

A flywheel is a mechanical device which uses the conservation of angular momentum to store rotational energy; a form of kinetic energy proportional to the product of its moment of inertia and the square of its rotational speed.

Tensor

Components stress tensor.svg (.

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

Tensors have become 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.

Aircraft principal axes

Free to rotate in three dimensions: yaw, nose left or right about an axis running up and down; pitch, nose up or down about an axis running from wing to wing; and roll, rotation about an axis running from nose to tail.

The position of all three axes, with the right-hand rule for describing the angle of its rotations
Yaw/heading, pitch and roll angles and associated vertical (down), transverse and longitudinal axes

These axes are related to the principal axes of inertia, but are not the same.

Angular momentum

Rotational analog of linear momentum.

This gyroscope remains upright while spinning due to the conservation of its angular momentum.
Velocity of the particle m with respect to the origin O can be resolved into components parallel to (v∥) and perpendicular to (v⊥) the radius vector r. The angular momentum of m is proportional to the perpendicular component v⊥ of the velocity, or equivalently, to the perpendicular distance r⊥ from the origin.
Relationship between force (F), torque (τ), momentum (p), and angular momentum (L) vectors in a rotating system. r is the position vector.
A figure skater in a spin uses conservation of angular momentum – decreasing her moment of inertia by drawing in her arms and legs increases her rotational speed.
The torque caused by the two opposing forces Fg and −Fg causes a change in the angular momentum L in the direction of that torque (since torque is the time derivative of angular momentum). This causes the top to precess.
The angular momentum of the particles i is the sum of the cross products R × MV + Σri × mivi.
The 3-angular momentum as a bivector (plane element) and axial vector, of a particle of mass m with instantaneous 3-position x and 3-momentum p.
Newton's derivation of the area law using geometric means.

Because angular momentum is the product of moment of inertia and angular velocity, if the angular momentum remains constant (is conserved), then the angular velocity (rotational speed) of the skater must increase.

Moment (mathematics)

In mathematics, the moments of a function are quantitative measures related to the shape of the function's graph.

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

If the function represents mass, then the first moment is the center of the mass, and the second moment is the rotational inertia.

Torque

Newton-metre .

Moment arm diagram
The torque caused by the two opposing forces Fg and −Fg causes a change in the angular momentum L in the direction of that torque. This causes the top to precess.
Torque curve of a motorcycle ("BMW K 1200 R 2005"). The horizontal axis shows the speed (in rpm) that the crankshaft is turning, and the vertical axis is the torque (in newton metres) that the engine is capable of providing at that speed.

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

Rigid body

Solid body in which deformation is zero or so small it can be neglected.

The position of a rigid body is determined by the position of its center of mass and by its attitude (at least six parameters in total).

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.

Rigid body dynamics

In the physical science of dynamics, rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces.

Movement of each of the components of the Boulton & Watt Steam Engine (1784) can be described by a set of equations of kinematics and kinetics
Tait–Bryan angles, another way to describe orientation.
Diagram of the Euler angles
Intrinsic rotation of a ball about a fixed axis.
Motion of a top in the Euler angles.

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

Moment (physics)

Mathematical expression involving the product of a distance and physical quantity.

A lever in balance

In 1765, the Latin term momentum inertiae (English: moment of inertia) is used by Leonhard Euler to refer to one of Christiaan Huygens's quantities in Horologium Oscillatorium.

Radius of gyration

Sphere rotating around one of its diameters

Radius of gyration or gyradius of a body about the axis of rotation is defined as the radial distance to a point which would have a moment of inertia the same as the body's actual distribution of mass, if the total mass of the body were concentrated there.