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The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a
rigid body
In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external fo ...
is a quantity that determines the
torque
In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
needed for a desired
angular acceleration about a rotational axis, akin to how
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
determines the
force needed for a desired
acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by t ...
. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rate of rotation.
It is an
extensive (additive) property: for a
point mass
A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take up ...
the moment of inertia is simply the mass times the square of the
perpendicular distance In geometry, the perpendicular distance between two objects is the distance from one to the other, measured along a line that is perpendicular to one or both.
The distance from a point to a line is the distance to the nearest point on that line. Th ...
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
An axis (plural ''axes'') is an imaginary line around which an object rotates or is symmetrical. Axis may also refer to:
Mathematics
* Axis of rotation: see rotation around a fixed axis
* Axis (mathematics), a designator for a Cartesian-coordinat ...
.
For bodies constrained to rotate in a plane, only their moment of inertia about an axis perpendicular to the plane, a
scalar value, matters. For bodies free to rotate in three dimensions, their moments can be described by a
symmetric 3 × 3 matrix, with a set of mutually perpendicular
principal axes for which this matrix is
diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δΠ...
and torques around the axes act independently of each other.
Introduction
When a body is free to rotate around an axis,
torque
In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
must be applied to change its
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
. 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·m
2) in
SI units and pound-foot-second squared (lbf·ft·s
2) in
imperial or
US units.
The moment of inertia plays the role in rotational kinetics that
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
(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
A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward th ...
.
The term ''moment of inertia'' was introduced by
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
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
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acc ...
, and in
Newton's laws of motion
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in moti ...
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
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. In particular, as ...
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
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...
of the section.
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The moment of inertia is also defined as the ratio of the net
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
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.
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
A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward th ...
, 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 ''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 concentr ...
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
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
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
![Mendenhall gravimeter pendulums](https://upload.wikimedia.org/wikipedia/commons/8/83/Mendenhall_gravimeter_pendulums.jpg)
A
compound pendulum
A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward th ...
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
Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are ...
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
(
) 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 The center of percussion is the point on an extended massive object attached to a pivot where a perpendicular impact will produce no reactive shock at the pivot. Translational and rotational motions cancel at the pivot when an impulsive blow is st ...
of the compound pendulum. This point also corresponds to the
center of percussion. The length
is determined from the formula,
or
The
seconds pendulum
A seconds pendulum is a pendulum whose period is precisely two seconds; one second for a swing in one direction and one second for the return swing, a frequency of 0.5 Hz.
Pendulum
A pendulum is a weight suspended from a pivot so that ...
, 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
A Kater's pendulum is a reversible free swinging pendulum invented by British physicist and army captain Henry Kater in 1817 for use as a gravimeter instrument to measure the local acceleration of gravity. Its advantage is that, unlike previous ...
is a compound pendulum that uses this property to measure the local acceleration of gravity, and is called a
gravimeter
Gravimetry is the measurement of the strength of a gravitational field. Gravimetry may be used when either the magnitude of a gravitational field or the properties of matter responsible for its creation are of interest.
Units of measurement
Gr ...
.
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 pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward th ...
. A trifilar pendulum is a platform supported by three wires designed to oscillate in torsion around its vertical centroidal axis. The period of oscillation of the trifilar pendulum yields the moment of inertia of the system.
Moment of inertia of area
Moment of inertia of area is also known as the
second moment of area
The second moment of area, or second area moment, or quadratic moment of area and also known as the area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. The ...
.
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 calculated thus
# Square:
# Rectangular:
and;
# Triangular:
# Circular:
Motion in a fixed plane
Point mass
![Rolling Racers - Moment of inertia](https://upload.wikimedia.org/wikipedia/commons/2/2e/Rolling_Racers_-_Moment_of_inertia.gif)
The moment of inertia about an axis of a body is calculated by summing
for every particle in the body, where
is the perpendicular distance to the specified axis. To see how moment of inertia arises in the study of the movement of an extended body, it is convenient to consider a rigid assembly of point masses. (This equation can be used for axes that are not principal axes provided that it is understood that this does not fully describe the moment of inertia.)
Consider the kinetic energy of an assembly of
masses
that lie at the distances
from the pivot point
, which is the nearest point on the axis of rotation. It is the sum of the kinetic energy of the individual masses,
[
]
This shows that the moment of inertia of the body is the sum of each of the
terms, that is
Thus, moment of inertia is a physical property that combines the mass and distribution of the particles around the rotation axis. Notice that rotation about different axes of the same body yield different moments of inertia.
The moment of inertia of a continuous body rotating about a specified axis is calculated in the same way, except with infinitely many point particles. Thus the limits of summation are removed, and the sum is written as follows:
Another expression replaces the summation with an
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
,
Here, the
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
gives the mass density at each point
,
is a vector perpendicular to the axis of rotation and extending from a point on the rotation axis to a point
in the solid, and the integration is evaluated over the volume
of the body
. The moment of inertia of a flat surface is similar with the mass density being replaced by its areal mass density with the integral evaluated over its area.
Note on second moment of area: The moment of inertia of a body moving in a plane and the
second moment of area
The second moment of area, or second area moment, or quadratic moment of area and also known as the area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. The ...
of a beam's cross-section are often confused. The moment of inertia of a body with the shape of the cross-section is the second moment of this area about the
-axis perpendicular to the cross-section, weighted by its density. This is also called the ''polar moment of the area'', and is the sum of the second moments about the
- and
-axes. The stresses in a
beam are calculated using the second moment of the cross-sectional area around either the
-axis or
-axis depending on the load.
Examples
![Moment of inertia rod center](https://upload.wikimedia.org/wikipedia/commons/b/b4/Moment_of_inertia_rod_center.svg)
The moment of inertia of a compound pendulum constructed from a thin disc mounted at the end of a thin rod that oscillates around a pivot at the other end of the rod, begins with the calculation of the moment of inertia of the thin rod and thin disc about their respective centers of mass.
* The moment of inertia of a thin rod with constant cross-section
and density
and with length
about a perpendicular axis through its center of mass is determined by integration.
Align the
-axis with the rod and locate the origin its center of mass at the center of the rod, then
where
is the mass of the rod.
* The moment of inertia of a thin disc of constant thickness
, radius
, and density
about an axis through its center and perpendicular to its face (parallel to its axis of
rotational symmetry
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...
) is determined by integration.
Align the
-axis with the axis of the disc and define a volume element as
, then
where
is its mass.
* The moment of inertia of the compound pendulum is now obtained by adding the moment of inertia of the rod and the disc around the pivot point
as,
where
is the length of the pendulum. Notice that the parallel axis theorem is used to shift the moment of inertia from the center of mass to the pivot point of the pendulum.
A
list of moments of inertia
Moment of inertia, denoted by , measures the extent to which an object resists rotational acceleration about a particular axis, it is the rotational analogue to mass (which determines an object's resistance to ''linear'' acceleration). The momen ...
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. The
parallel axis theorem
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 moment of inertia or the second moment of area of a rigid body abou ...
is used to shift the reference point of the individual bodies to the reference point of the assembly.
![Moment of inertia solid sphere](https://upload.wikimedia.org/wikipedia/commons/1/19/Moment_of_inertia_solid_sphere.svg)
As one more example, consider the moment of inertia of a solid sphere of constant density about an axis through its center of mass. This is determined by summing the moments of inertia of the thin discs that can form the sphere whose centers are along the axis chosen for consideration. If the surface of the ball is defined by the equation
then the square of the radius
of the disc at the cross-section
along the
-axis is
Therefore, the moment of inertia of the ball is the sum of the moments of inertia of the discs along the
-axis,
where
is the mass of the sphere.
Rigid body
![RollingVsInertia](https://upload.wikimedia.org/wikipedia/commons/9/95/RollingVsInertia.gif)
If a
mechanical system is constrained to move parallel to a fixed plane, then the rotation of a body in the system occurs around an axis
perpendicular to this plane. In this case, the moment of inertia of the mass in this system is a scalar known as the ''polar moment of inertia''. The definition of the polar moment of inertia can be obtained by considering momentum, kinetic energy and Newton's laws for the planar movement of a rigid system of particles.
If a system of
particles,
, are assembled into a rigid body, then the momentum of the system can be written in terms of positions relative to a reference point
, and absolute velocities
:
where
is the angular velocity of the system and
is the velocity of
.
For planar movement the angular velocity vector is directed along the unit vector
which is perpendicular to the plane of movement. Introduce the unit vectors
from the reference point
to a point
, and the unit vector
, so
This defines the relative position vector and the velocity vector for the rigid system of the particles moving in a plane.
Note on the cross product: When a body moves parallel to a ground plane, the trajectories of all the points in the body lie in planes parallel to this ground plane. This means that any rotation that the body undergoes must be around an axis perpendicular to this plane. Planar movement is often presented as projected onto this ground plane so that the axis of rotation appears as a point. In this case, the angular velocity and angular acceleration of the body are scalars and the fact that they are vectors along the rotation axis is ignored. This is usually preferred for introductions to the topic. But in the case of moment of inertia, the combination of mass and geometry benefits from the geometric properties of the cross product. For this reason, in this section on planar movement the angular velocity and accelerations of the body are vectors perpendicular to the ground plane, and the cross product operations are the same as used for the study of spatial rigid body movement.
Angular momentum
The angular momentum vector for the planar movement of a rigid system of particles is given by
Use the
center of mass as the reference point so
and define the moment of inertia relative to the center of mass
as
then the equation for angular momentum simplifies to
The moment of inertia
about an axis perpendicular to the movement of the rigid system and through the center of mass is known as the ''polar moment of inertia''. Specifically, it is the
second moment of mass with respect to the orthogonal distance from an axis (or pole).
For a given amount of angular momentum, a decrease in the moment of inertia results in an increase in the angular velocity. Figure skaters can change their moment of inertia by pulling in their arms. Thus, the angular velocity achieved by a skater with outstretched arms results in a greater angular velocity when the arms are pulled in, because of the reduced moment of inertia. A figure skater is not, however, a rigid body.
Kinetic energy
![Lever shear flywheel](https://upload.wikimedia.org/wikipedia/commons/f/fe/Lever_shear_flywheel.jpg)
The kinetic energy of a rigid system of particles moving in the plane is given by
Let the reference point be the center of mass
of the system so the second term becomes zero, and introduce the moment of inertia
so the kinetic energy is given by
The moment of inertia
is the ''polar moment of inertia'' of the body.
Newton's laws
![Johndeered](https://upload.wikimedia.org/wikipedia/commons/c/c9/Johndeered.jpg)
Newton's laws for a rigid system of
particles,
, can be written in terms of a
resultant force
In physics and engineering, a resultant force is the single force and associated torque obtained by combining a system of forces and torques acting on a rigid body via vector addition. The defining feature of a resultant force, or resultant for ...
and torque at a reference point
, to yield
where
denotes the trajectory of each particle.
The
kinematics of a rigid body yields the formula for the acceleration of the particle
in terms of the position
and acceleration
of the reference particle as well as the angular velocity vector
and angular acceleration vector
of the rigid system of particles as,
For systems that are constrained to planar movement, the angular velocity and angular acceleration vectors are directed along
perpendicular to the plane of movement, which simplifies this acceleration equation. In this case, the acceleration vectors can be simplified by introducing the unit vectors
from the reference point
to a point
and the unit vectors
, so
This yields the resultant torque on the system as
where
, and
is the unit vector perpendicular to the plane for all of the particles
.
Use the
center of mass as the reference point and define the moment of inertia relative to the center of mass
, then the equation for the resultant torque simplifies to
Motion in space of a rigid body, and the inertia matrix
The scalar moments of inertia appear as elements in a matrix when a system of particles is assembled into a rigid body that moves in three-dimensional space. This inertia matrix appears in the calculation of the angular momentum, kinetic energy and resultant torque of the rigid system of particles.
[L. W. Tsai, Robot Analysis: The mechanics of serial and parallel manipulators, John-Wiley, NY, 1999.]
Let the system of
particles,
be located at the coordinates
with velocities
relative to a fixed reference frame. For a (possibly moving) reference point
, the relative positions are
and the (absolute) velocities are
where
is the angular velocity of the system, and
is the velocity of
.
Angular momentum
Note that the
cross product can be equivalently written as matrix multiplication by combining the first operand and the operator into a skew-symmetric matrix,