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Moment of inertia

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This article is about the moment of inertia of a rotating object. For the moment of inertia dealing with bending of a plane, see second moment of area.
Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI units kg m2, English units ft lb s2) quantifies the rotational inertia of a rigid body, i.e. its inertia with respect to rotational motion, in a manner somewhat analogous to how mass quantifies the inertia of a body with respect to translational motion. The symbols [I] and sometimes [J] are usually used to refer to the moment of inertia.

The moment of inertia of an object about a given axis describes how difficult it is to induce an angular rotation of the object about that axis. For example, consider two wheels of the same mass, one large and one small in radius. The smaller wheel is easier to accelerate into spinning fast, because its mass is concentrated close to the axis of rotation. Conversely, the larger wheel takes more effort to accelerate into spinning fast, because its mass is spread out further from the axis of rotation. Quantitatively, the smaller wheel has a smaller moment of inertia, whereas the larger wheel has a larger moment of inertia.

The moment of inertia has two forms, a scalar form [I] (used when the axis of rotation [\mathbf}] is known) and a more general tensor form [\mathbf] that does not require knowing the axis of rotation. The scalar form [I] for any axis can be calculated from the tensor form [\mathbf] using the double dot product

[I = \mathbf} \cdot \mathbf \cdot \mathbf} = \sum_^ \sum_^ n_ I_ n_]
where the summations are over the three Cartesian coordinates. The scalar moment of inertia [I] is often called simply the "moment of inertia".

The moment of inertia is sometimes called the mass moment of inertia (especially by mechanical engineers) to avoid confusion with the second moment of area, which is sometimes called the moment of inertia (especially by structural engineers) and denoted by the same symbol [I].

Contents

Definition of the (scalar) moment of inertia

The (scalar) moment of inertia of a point mass rotating about a known axis is defined by

[I \equiv m r^2\,]
where
m is its mass,
and r is its perpendicular distance from the axis of rotation.
The moment of inertia is additive so, for a rigid body consisting of [N] point masses [m_] with distances [r_] to the rotation axis, the total moment of inertia equals the sum of the point-mass moments of inertia

[I \equiv \sum_^ r_^2}]
Generalizing to a solid body described by a continuous mass-density function [\rho(\mathbf)], the moment of inertia for rotating about a known axis can be calculated by integrating the moments of the masses relative to the rotation axis

[I \equiv \int_V r^2\,dm = \iiint_V = \iiint_V \!]
where
V is the volume of the object
r² dm is the infinitesimal moment of inertia
r is the distance from the axis of rotation
dm is an infinitesimal mass
ρ is the mass density of the object
dv is an infinitesimal volume
and dx, dy, and dz are infinitesimal lengths.

Parallel axis theorem

If the moment of inertia has been calculated for rotations about the centroid of a rigid body, we can conveniently calculate the moment of inertia for all parallel rotations as well (without having to resort to the formal definition given above). If the axis of rotation is displaced by a distance [R] from the centroid axis of rotation (e.g., spinning a disc about a point on its periphery, rather than through its center), the new moment of inertia equals:

[ I^} = I^} + M R^ \ ]
where
[M] is the total mass of the rigid body, and
R is the distance of the axis of rotation from the centroid axis of rotation (as described above).
This theorem is also known as parallel axes rule or Steiner's theorem.

As an aside, the formal definition above is always correct, whatever the axis of rotation is.

Kinetic energy

For a system with [N] point masses [m_] moving with speeds [v_], the kinetic energy [T] always equals

[T = \sum_^ \frac m_ v_^]
For a rigid body rotating with angular speed [\omega], the speeds can be written

[v_ = \omega r_]
where again [r_] is the shortest distance from the point mass to the rotation axis. Therefore, the kinetic energy can be written

[T = \sum_^ \frac m_ \omega^ r_^ = \frac I \omega^]
The final formula [T=\frac I \omega^] also holds for a continuous distribution of mass.

Angular momentum and torque

Similarly, the angular momentum [\mathbf] for a system of particles is defined

[\mathbf = \sum_^ \mathbf_ \times \mathbf_ = \sum_^ m_ \mathbf_ \times \mathbf_]
For a rigid body rotating with angular velocity [\omega] about the rotation axis [\mathbf}] (a unit vector), the velocity vector [\mathbf_] may be written as a vector cross product

[\mathbf_ = \omega \mathbf} \times \mathbf_ \equiv \boldsymbol\omega \times \mathbf_]
where we have introduced the angular velocity vector [\boldsymbol\omega \equiv \omega \mathbf}] and [\mathbf_] is the shortest vector from the rotation axis to the point mass. Substituting the formula for [\mathbf_] into the definition of [\mathbf] yields

[\mathbf = \sum_^ m_ \mathbf_ \times \boldsymbol\omega \times \mathbf_ = \boldsymbol\omega \sum_^ m_ r_^ = I \omega \mathbf}]
where we have used the fact that [\boldsymbol\omega \cdot \mathbf_ = 0]. The torque [\mathbf] is defined as the rate of change of the angular momentum [\mathbf]

[\mathbf \equiv \frac}]
If the torque is driving the rotation in the same direction [\mathbf}] (so that neither I nor [\mathbf}] are changing) then we may write

[\mathbf \equiv I \frac\mathbf} = I \alpha \mathbf}]
where [\alpha] is called the angular acceleration (or rotational acceleration) about the rotation axis [\mathbf}].

Conservation of angular momentum explains the speeding up of ice skaters when they pull in their arms. Since the ice is nearly frictionless, the angular momentum should stay constant during their spin. When they pull in their arms, the skaters decrease their average radius (their mass is more concentrated close to the rotation axis). To keep the angular momentum constant, the angular velocity [\omega] must increase; hence, the skaters spin faster. Similarly, divers spin faster when they're tightly curled in a ball (small radius), and spin more slowly when they extend their bodies (large radius) just before hitting the water. Similar phenomena are also seen with gymnasts.

Moment of inertia tensor

For the same object, different axes of rotation will have different moments of inertia about that axis. For example, the three moments of inertia associated with rotations about the three Cartesian axes (X, Y, and Z) are not guaranteed to be equal unless the object is very symmetric

[I_ = \;] moment of inertia about the line through the centroid, parallel to the X-axis
[I_ = \;] moment of inertia about the line through the centroid, parallel to the Y-axis
[I_ = \;] moment of inertia about the line through the centroid, parallel to the Z-axis
These elements are part of a more general moment of inertia tensor [\mathbf] whose components are defined

[I_ \equiv \sum_^ m_ (y_^+z_^)]
[I_ \equiv \sum_^ m_ (x_^+z_^)]
[I_ \equiv \sum_^ m_ (x_^+y_^)]
[I_ = I_ \equiv -\sum_^ m_ x_ y_\;]
[I_ = I_ \equiv -\sum_^ m_ x_ z_\;]
[I_ = I_ \equiv -\sum_^ m_ y_ z_\;]
for a system with [N] point masses [m_] with Cartesian coordinates [(x_,y_,z_)]. Since this tensor is a symmetric, real matrix, it is possible to find a Cartesian coordinate system in which it is diagonal, i.e., has the form

[\mathbf = \beginI_ & 0 & 0 \\0 & I_ & 0 \\0 & 0 & I_\end]
where the coordinate axes are called the principal axes and the constants [I_], [I_] and [I_] are called the principal moments of inertia and are usually arranged in increasing order

[I_ \leq I_ \leq I_]
The unit vectors along the principal axes are usually denoted as [(\mathbf_, \mathbf_, \mathbf_)].

Symmetry axes and the inertia tensor

If two principal moments are the same, the rigid body is called a symmetrical top and there is no unique choice for the two corresponding principal axes. If all three principal moments are the same, the rigid body is called a spherical top (although it may not be spherical) and any axis can be considered a principal axis (since all are equivalent).

The principal axes are often aligned with the object's symmetry axes. If a rigid body has an axis of symmetry of order [m], i.e., is symmetrical under rotations of [\frac}] about a given axis, the symmetry axis is a principal axis. If [m>2], the rigid body is a symmetrical top. If a rigid body has at least two symmetry axes that are not parallel or perpendicular to each other, it is a spherical top, e.g., a cube or any other Platonic solid.

Parallel axes theorem in tensor form

If the moment of inertia tensor has been calculated for rotations about the centroid of the rigid body, it is relatively easy to compute the tensor for rotations offset from the centroid. If the axis of rotation is displaced by a vector [\mathbf] from the centroid, the new moment of inertia tensor equals

[I^}_ = I^}_ + M \left[ R^ delta_ - R_ R_ right]]
where [M] is the total mass of the rigid body, [R^ \equiv \mathbf \cdot \mathbf] and [\delta_] is the Kronecker delta function. Of course, the new moment of inertia tensor could have been computed directly using the formal definitions above; this is merely a useful labor-saving method.

Mechanical quantities expressed using the tensor

Using the tensor [\mathbf], the kinetic energy can be written as a double dot product

[T = \frac \boldsymbol\omega \cdot \mathbf \cdot \boldsymbol\omega = \frac I_ \omega_^ + \frac I_ \omega_^ + \frac I_ \omega_^]
and the angular moment can be written as a single dot product

[\mathbf = \mathbf \cdot \boldsymbol\omega = \omega_ I_ \mathbf_ + \omega_ I_ \mathbf_ + \omega_ I_ \mathbf_]
Taken together, we may express the kinetic energy in terms of the angular momentum [(L_, L_, L_)] in the principal axis frame

[T = \frac^}} + \frac^}} + \frac^}}]
where [L_ \equiv I_ \omega_] for [k=1,2,3].

See also

References

External links

 

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