Parallel axis theorem

In physics

Physics

Physics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...

, the parallel axis theorem or Huygens-Steiner theorem can be used to determine the moment of inertia

Moment of inertia

Moment of inertia, also called mass moment of inertia or the angular mass, is a measure of an object's resistance to changes in its rotation rate. It is the rotational analog of mass. That is, it is the inertia of a rigid rotating body with respect to its rotation...

 of a rigid body

Rigid body

In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...

 about any axis, given the moment of inertia of the object about the parallel

Parallel (geometry)

Parallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not intersect or...

 axis through the object's center of mass

Center of mass

The center of mass of a system of particles is a specific point where, for many purposes, the system behaves as if its mass were concentrated there. The center of mass is a function only of the positions and masses of the particles that compose the system...

 and the perpendicular

Perpendicular

In geometry, two lines or planes , are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective. Thus, referring to Figure 1, the line AB is the perpendicular to CD through the point B...

 distance

Distance

Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, a period of time, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical...

 between the axes.

Let:

I_cm denote the moment of inertia of the object about its center of mass,

M the object's mass and d the perpendicular distance between the two axes.

In physics

Physics

Physics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force...

, the parallel axis theorem or Huygens-Steiner theorem can be used to determine the moment of inertia

Moment of inertia

Moment of inertia, also called mass moment of inertia or the angular mass, is a measure of an object's resistance to changes in its rotation rate. It is the rotational analog of mass. That is, it is the inertia of a rigid rotating body with respect to its rotation...

 of a rigid body

Rigid body

In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...

 about any axis, given the moment of inertia of the object about the parallel

Parallel (geometry)

Parallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not intersect or...

 axis through the object's center of mass

Center of mass

The center of mass of a system of particles is a specific point where, for many purposes, the system behaves as if its mass were concentrated there. The center of mass is a function only of the positions and masses of the particles that compose the system...

 and the perpendicular

Perpendicular

In geometry, two lines or planes , are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective. Thus, referring to Figure 1, the line AB is the perpendicular to CD through the point B...

 distance

Distance

Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, a period of time, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical...

 between the axes.

Let:

I_cm denote the moment of inertia of the object about its center of mass,

M the object's mass and d the perpendicular distance between the two axes.


Then the moment of inertia about the new axis z is given by:
This rule can be applied with the stretch rule

Stretch rule

In physics, the stretch rule states that the moment of inertia of a rigid object is unchanged when the object is stretched parallel to the axis of rotation, In physics, the stretch rule states that the moment of inertia of a rigid object is unchanged when the object is stretched parallel to the...

 and perpendicular axis theorem

Perpendicular axis theorem

In physics, the perpendicular axis theorem can be used to determine the moment of inertia of a rigid object that lies entirely within a plane, about an axis at right angles to the plane, given the moments of inertia of the object about two perpendicular axes lying within the plane...

 to find moments of inertia for a variety of shapes.


The parallel axes rule also applies to the second moment of area

Second moment of area

The second moment of area, also known as the area moment of inertia or second moment of inertia is a property of a shape that can be used to predict the resistance of beams to bending and deflection. The deflection of a beam under load depends not only on the load, but also on the geometry of the...

 (area moment of inertia)
for a plane region D:

In this formula,

I_z is the area moment of inertia of D relative to the parallel axis,

I_x is the area moment of inertia of D relative to its centroid

Centroid

In geometry, the centroid, geometric center, or barycenter of a plane figure X is the intersection of all straight lines that divide X into two parts of equal moment about the line. Informally, it is the "average" of all points of X...

,

A is the area of the plane region D, and

d is the distance from the new axis z to the centroid

Centroid

In geometry, the centroid, geometric center, or barycenter of a plane figure X is the intersection of all straight lines that divide X into two parts of equal moment about the line. Informally, it is the "average" of all points of X...

 of the plane region D.

Note: The centroid

Centroid

In geometry, the centroid, geometric center, or barycenter of a plane figure X is the intersection of all straight lines that divide X into two parts of equal moment about the line. Informally, it is the "average" of all points of X...

 of D coincides with the center of gravity (CG) of a physical plate
with the same shape that has constant density.

In classical mechanics

In classical mechanics, the Parallel axis theorem (also known as Huygens-Steiner theorem) can be generalized to calculate a new inertia tensor J_ij from an inertia tensor about a center of mass I_ij when the pivot point is a displacement a from the center of mass:
where
is the displacement vector from the center of mass to the new axis, and
is the Kronecker delta

Kronecker delta

In mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker , is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise...

.

We can see that, for diagonal elements (when i = j), displacements perpendicular to the axis of rotation results in the above simplified version of the parallel axis theorem.