The moment of inertia (also known as mass moment of inertia, angular/rotational mass, second moment of mass, or rotational inertia) is a measure of how difficult it is to change the rotation rate of a rigid body about a given axis. It is the ratio between the torque applied and the resulting angular acceleration about that axis. It plays the same role in rotational motion as mass does in linear motion. A body's moment of inertia about a particular axis depends on both the mass and its distribution relative to the axis, increasing with mass and distance from the axis.
For a point mass, the moment of inertia is simply the mass times the square of the perpendicular distance to the axis of rotation. The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis). Its simplest definition is the second moment of mass with respect to distance from an axis.
If a body is forced to rotate in a plane, then only the moment of inertia about an axis perpendicular to the plane, a scalar value, matters. If a body is allowed to rotate in all three dimensions, then its moment is described by a symmetric three-by-three matrix. This symmetric matrix can be orthogonally diagonalized by a set of mutually perpendicular principal axes. Torques around these principal axes act independently of each other.
Introduction When a body is free to rotate around an axis, torque must be applied to change its angular momentum. The amount of torque needed to cause any given angular acceleration (the rate of change in angular velocity) is proportional to the moment of inertia of the body. Moments of inertia may be expressed in units of kilogram metre squared (kg·m2) in SI units and pound-foot-squared (lb·ft2) in imperial or US units, and other relevant units.
The moment of inertia plays the role in rotational kinetics that mass (inertia) plays in linear kinetics—both characterize the resistance of a body to changes in its motion. The moment of inertia depends on how mass is distributed around an axis of rotation, and will vary depending on the chosen axis. For a point-like mass, the moment of inertia about some axis is given by , where is the distance of the point from the axis, and is the mass. For an extended rigid body, the moment of inertia is just the sum of all the small pieces of mass multiplied by the square of their distances from the axis in rotation. For an extended body of a regular shape and uniform density, this summation sometimes produces a simple expression that depends on the dimensions, shape and total mass of the object.
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. The term moment of inertia ("momentum inertiae" in Latin) was introduced by Leonhard Euler in his book Theoria motus corporum solidorum seu rigidorum in 1765, From page 166: "Definitio 7. 422. Momentum inertiae corporis respectu eujuspiam axis est summa omnium productorum, quae oriuntur, si singula corporis elementa per quadrata distantiarum suarum ab axe multiplicentur." (Definition 7. 422. A body's moment of inertia with respect to any axis is the sum of all of the products, which arise, if the individual elements of the body are multiplied by the square of their distances from the axis.) and it is incorporated into Euler's second law.
The natural frequency of oscillation of a compound pendulum is obtained from the ratio of the torque imposed by gravity on the mass of the pendulum to the resistance to acceleration defined by the moment of inertia. Comparison of this natural frequency to that of a simple pendulum consisting of a single point of mass provides a mathematical formulation for moment of inertia of an extended body.
The 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. There is an interesting difference in the way moment of inertia appears in planar and spatial movement. Planar movement has a single scalar that defines the moment of inertia, while for spatial movement the same calculations yield a 3 × 3 matrix of moments of inertia, called the inertia matrix or inertia tensor.
The moment of inertia of a rotating flywheel is used in a machine to resist variations in applied torque to smooth its rotational output. The moment of inertia of an airplane about its longitudinal, horizontal and vertical axes determine how steering forces on the control surfaces of its wings, elevators and rudder(s) affect the plane's motions in roll, pitch and yaw.
Definition The moment of inertia is defined as the product of mass of section and the square of the distance between the reference axis and the centroid of the section. It is denoted with the symbol I or J.
The moment of inertia is also defined as the ratio of the net angular momentum of a system to its angular velocity around a principal axis, that is
If the angular momentum of a system is constant, then as the moment of inertia gets smaller, the angular velocity must increase. This occurs when spinning figure skaters pull in their outstretched arms or divers curl their bodies into a tuck position during a dive, to spin faster.
If the shape of the body does not change, then its moment of inertia appears in Newton's law of motion as the ratio of an applied torque on a body to the angular acceleration around a principal axis, that is
For a simple pendulum, this definition yields a formula for the moment of inertia in terms of the mass of the pendulum and its distance from the pivot point as,
Thus, the moment of inertia of the pendulum depends on both the mass of a body and its geometry, or shape, as defined by the distance to the axis of rotation.
This simple formula generalizes to define moment of inertia for an arbitrarily shaped body as the sum of all the elemental point masses each multiplied by the square of its perpendicular distance to an axis . An arbitrary object's moment of inertia thus depends on the spatial distribution of its mass.
In general, given an object of mass , an effective radius can be defined, dependent on a particular axis of rotation, with such a value that its moment of inertia around the axis is
where is known as the radius of gyration around the axis.
Examples
Simple pendulum Mathematically, the moment of inertia of a simple pendulum is the ratio of the torque due to gravity about the pivot of a pendulum to its angular acceleration about that pivot point. For a simple pendulum, this is found to be the product of the mass of the particle with the square of its distance to the pivot, that is
This can be shown as follows:
The force of gravity on the mass of a simple pendulum generates a torque around the axis perpendicular to the plane of the pendulum movement. Here is the distance vector from the torque axis to the pendulum center of mass, and is the net force on the mass. Associated with this torque is an angular acceleration, , of the string and mass around this axis. Since the mass is constrained to a circle the tangential acceleration of the mass is . Since the torque equation becomes:
where is a unit vector perpendicular to the plane of the pendulum. (The second to last step uses the vector triple product expansion with the perpendicularity of and .) The quantity is the moment of inertia of this single mass around the pivot point.
The quantity also appears in the angular momentum of a simple pendulum, which is calculated from the velocity of the pendulum mass around the pivot, where is the angular velocity of the mass about the pivot point. This angular momentum is given by
using a similar derivation to the previous equation.
Similarly, the kinetic energy of the pendulum mass is defined by the velocity of the pendulum around the pivot to yield
This shows that the quantity is how mass combines with the shape of a body to define rotational inertia. The moment of inertia of an arbitrarily shaped body is the sum of the values for all of the elements of mass in the body.
Compound pendulums
A compound pendulum is a body formed from an assembly of particles of continuous shape that rotates rigidly around a pivot. Its moment of inertia is the sum of the moments of inertia of each of the particles that it is composed of. The natural frequency () of a compound pendulum depends on its moment of inertia, ,
where is the mass of the object, is local acceleration of gravity, and is the distance from the pivot point to the center of mass of the object. Measuring this frequency of oscillation over small angular displacements provides an effective way of measuring moment of inertia of a body.
Thus, to determine the moment of inertia of the body, simply suspend it from a convenient pivot point so that it swings freely in a plane perpendicular to the direction of the desired moment of inertia, then measure its natural frequency or period of oscillation (), to obtain
where is the period (duration) of oscillation (usually averaged over multiple periods).
Center of oscillation A simple pendulum that has the same natural frequency as a compound pendulum defines the length from the pivot to a point called the center of oscillation of the compound pendulum. This point also corresponds to the center of percussion. The length is determined from the formula,
The seconds pendulum, which provides the "tick" and "tock" of a grandfather clock, takes one second to swing from side-to-side. This is a period of two seconds, or a natural frequency of for the pendulum. In this case, the distance to the center of oscillation, , can be computed to be
Notice that the distance to the center of oscillation of the seconds pendulum must be adjusted to accommodate different values for the local acceleration of gravity. Kater's pendulum is a compound pendulum that uses this property to measure the local acceleration of gravity, and is called a gravimeter.
Measuring moment of inertia The moment of inertia of a complex system such as a vehicle or airplane around its vertical axis can be measured by suspending the system from three points to form a trifilar pendulum. A trifilar pendulum is a platform supported by three wires designed to oscillate in torsion around its vertical centroidal axis.C. Couch and J. Mayes, Trifilar Pendulum for MOI, Happresearch.com, 2016. The period of oscillation of the trifilar pendulum yields the moment of inertia of the system.Gracey, William, The experimental determination of the moments of inertia of airplanes by a simplified compound-pendulum method, NACA Technical Note No. 1629, 1948
Moment of inertia of area Moment of inertia of area is also known as the second moment of area and its physical meaning is completely different from the mass moment of inertia.
These calculations are commonly used in civil engineering for structural design of beams and columns. Cross-sectional areas calculated for vertical moment of the x-axis and horizontal moment of the y-axis .
Height (h) and breadth (b) are the linear measures, except for circles, which are effectively half-breadth derived,
Sectional areas moment formulae: Square: Rectangular: and; Triangular: Circular:
Motion in a fixed plane Point mass