Moment of inertia
The moment of inertia, and mass moment of inertia or Inertialmoment is the resistance of a rigid body with respect to a change in its rotational motion ( divided by angular acceleration torque) around a given axis. Thus, it plays the same role as in the relationship between force and acceleration, the mass; therefore is in the older literature the term rotational mass use. As the physical size it is the first time in 1740 before the factory Theoria motus corporum solidorum seu rigidorum by Leonhard Euler.
The moment of inertia depends on the mass distribution with respect to the axis of rotation. The further a mass element remote from the axis of rotation, the more it contributes to the moment of inertia; the distance is a square.
If the rotation axis not fixed, so not enough to describe the inertial behavior of a single number. From the inertia tensor of the moment of inertia for any axis through the center of gravity can be calculated.
- 5.1 Rigid body consisting of mass points
- 5.2 Rigid body mass distribution described by
- 7.1 Homogeneous mass distribution
- 7.2 Moment of inertia of rotationally symmetric body
- 7.3 Moment of inertia with respect to each other parallel axes
- 8.1 Rotation of the coordinate system 8.1.1 Example calculation: Rotational body
- 9.1 principal moment of inertia
- 9.2 Moment of inertia for clamped axis
Illustrative examples
Balancierhilfe
When Seiltanz be used as Balancierhilfe preferred long poles. Compared to an equal weight compact body, such as a sandbag, a rod has a very large moment of inertia. A to -side tipping is not thereby prevented, but slowed down so that the Artist has enough time for a balancing movement.
The effect you can easily try it yourself: A 30 -cm ruler ( shorter is more difficult ) can balance upright on the palm. Cross, however, placed on one of its long edges, it falls to complete before you can respond. The axis of rotation is in both cases the overlying edge, while the mean square distance from this axis, with over 300 cm2, or approximately 4 cm2 is greatly different.
The short distance quadratically in the moment of inertia can be easily view: A given angular acceleration means for a mass element in double spacing twice the tangential acceleration and thus twice as large inertia force. The torque, twice the force × lever arm twice, so that is four times as large.
Swivel chair and Pirouette
With another simple experiment can illustrate a change in the moment of inertia. You sit as centrally as possible on a rotating office chair and can be offset with stretched arms and legs in rotation. If you then zoom pulls the arms and legs to the body, the moment of inertia decreases. The result is that the rotational movement is faster, because the angular momentum is conserved (see conservation of angular momentum ). Re- stretching slows down the movement. To enhance the effect, you can take in each hand heavy objects, such as dumbbells. The greater the mass, the more the effect.
A similar example is the pirouette effect, which is known from the figure skating. The control of the speed of rotation can be done only from the transfer of body mass relative to the axis of rotation. Taking the figure skaters arms at or directed from a squatting position straight so it rotates faster - a renewed momentum overtaking is not necessary.
Symbols and unit
The most common symbol for the moment of inertia and, going back to the Latin word iners, meaning idle and sluggish. However, since both symbols are also used in electrical engineering, a (large theta) is still being widely used. This article is used throughout.
The SI unit of moment of inertia is kg · m2.
Comparison with the mass for linear motion
The moment of inertia for a rotating movement is comparable to the mass of a linear ( straight-line ) motion (for details see Rotation ( physics ) # comparison with the translational motion). Compare the following equations:
General definition
The moment of inertia can be at a known mass distribution of a body of the following volume integral calculate:
Here, the axis of rotation ( angular velocity) is vertical portion of ( see illustration ).
Motivation of the definition
Rigid body consisting of mass points
The total kinetic energy of a rigid body, which consists of mass points, results from the sum of the kinetic energies of the individual mass points:
Here, the web speed of the i- th material point. Now, the entire body is rotated about said axis. Every single mass point, therefore, describes a circular path. The web speed of a particle that rotates on a circular path with radius with the angular velocity, can be calculated as. Hence it follows:
Analogous to the definition of the kinetic energy
A linearly moving rigid body consisting of N point masses with the total mass, we define the moment of inertia of a rotating rigid body consisting of N point masses as
It is therefore
By this definition, one can identify the following sizes of rotating mass points with the linear sizes of moving mass points:
Choose the z-axis of the coordinate system in the direction of the axis of rotation, can be derived as the following practical equation:
And wherein x and y coordinates of the i- th material point in the coordinate system are selected so. The subscript " z" is important because the moment of inertia of a body on a rotation axis ( here the z-axis ) is always relative. The equation can also be seen that the moment of inertia is not dependent on the z coordinates of each mass point. The moment of inertia is independent of the coordinates of the mass points in the direction of the rotation axis.
Rigid body mass distribution described by
The formula for the moment of inertia of a general mass distribution is obtained in which you built up the mass distribution of many small mass elements, imagines. The rotational energy is then given by
Given. The transition to the integral with the volume of the composite from the infinitesimal mass elements body results,
From this, the above mentioned general definition of the moment of inertia with a position-dependent results (ie generally inhomogeneous ) mass density.
Context of the moment of inertia with angular momentum
The total angular momentum of the rigid body does not show A., in the same direction as the angular velocity. The axis- parallel component, however, is given by. This can be see as follows. The position vector of a single mass element is according to parallel in a distributed and a perpendicular component. To achsenparallen component of the angular momentum of this mass element of the parallel component of the position vector contributes nothing, it remains:
The axis- parallel component of the total angular momentum is then given by
Moreover, it follows immediately
Formulas for important special cases
Homogeneous distribution of mass
In a homogeneous mass distribution, the density is locally constant. The density can be pulled out of the integral and the formula for the moment of inertia simplifies to
Below is a sample calculation is given.
Moment of inertia of rotationally symmetric body
The moment of inertia of rotationally symmetrical bodies, which rotate about their axis of symmetry ( z-axis) can be easily calculated with the aid of cylinder coordinates. This must either be known, the height as a function of the radius ( ) and the radius as a function of the z- coordinate ( ). The volume element in cylindrical coordinates is given by. The integrations over and over and and are easy to perform and yields:
Moment of inertia with respect to each other parallel axes
If the moment of inertia for an axis through the center of gravity of a body is known, can be calculated using the " Steiner's phrase" the moment of inertia for an arbitrary parallel-displaced axis of rotation. The formula is:
This gives the distance between the axis through the center of gravity shifted parallel to the axis of rotation.
One can generalize the Steiner's theorem for any two parallel axes of rotation. For this, the set must be applied twice: first you move the axis of rotation so that it goes through the center of gravity of the body, then to the desired destination.
Generalization by inertia tensor
The inertia tensor of a body is a generalization of the moment of inertia. In a Cartesian coordinate system, the inertia tensor can be represented as a matrix, which is composed of the moments of inertia with respect to the three coordinate axes, and the products of inertia. The three moments of inertia are the diagonal of the matrix of inertia are the off-diagonal elements. With the help of the inertia tensor can be, for example, the moment of inertia with respect to an arbitrary continuous through the center axis directions. If a rigid body to such an axis rotates with the angular velocity, so the moment of inertia results to
Or in matrix notation
Rotation of the coordinate system
One axis in any direction in space is described by the unit vector. This can be obtained, for example, by rotating the unit vector in the z direction by means of a rotation matrix R:
With one obtains
By means of this rotation matrix can now be transformed into a coordinate system of the inertia tensor, in which the z- axis points in the direction of the axis of rotation:
The moment of inertia for the new z- axis is now just the third diagonal element of the tensor in the new representation. After execution of the matrix multiplication and trigonometric transformations resulting
Example calculation: Rotational body
We consider as an example to the inertia tensor of a rotationally symmetrical body. If one of the coordinate axes coincide (in this case the z-axis ) to the axis of symmetry, then this is diagonal tensor. The moments of inertia for rotation about the x-axis and the y-axis are equal to (). For the z axis, the moment of inertia can be different ( ). The inertia tensor thus has the following form:
Transforming these tensor as described above in a coordinate system which is rotated by the angle about the y- axis, we get:
This results in:
Special moments of inertia
Principal moment of inertia
Looking at an irregularly shaped body which rotates about an axis through its center of gravity, its moment of inertia varies depending on the location of the axis of rotation. There are two axes with respect to which the moment of inertia of the body, maximum and minimum. These axes are always perpendicular to each other and, together with a third, again perpendicular to both axis the principal axes of inertia of the body. In a plane spanned by the principal axes coordinate system of the inertia tensor is diagonal. Belonging to the principal axes of inertia moments of inertia are thus the eigenvalues of the inertia tensor, they are called principal moments of inertia. The corresponding coordinate system is called the principal inertial system.
The principal axes of inertia coincide with any existing axes of symmetry of the body. If two principal moments of inertia equal, all axes of rotation in the plane spanned by the corresponding principal axes of inertia, also principal axes of inertia of the same moment of inertia. This is immediately clear for cylindrically symmetric bodies, but also applies, for example for a rod with a square or hexagonal basis. In the event that all principal moments of inertia are identical, as was shown above, each axis of rotation through the center of a principal axis of inertia with the same moment of inertia. For all regular solids such as ball, equilateral tetrahedron, cube, etc. Accordingly, the moment of inertia for each axis through the center of gravity is the same.
See also: momental
Moment of inertia for the clamped axis
When a rigid body rotates about a fixed axis which is clamped at an angular velocity (the direction of the vector is the direction of the rotational axis), so can be the angular momentum of the general formula to calculate. Here, in contrast to the formula given above is not the moment of inertia, but the inertia tensor. In general, the angular momentum has now is not the direction of the rotation axis and is not constant over time, so that the bearings need to constantly apply torques (dynamic imbalance). Is only for rotation about one of the principal axes of inertia.
Applies to the angular momentum component along the axis of rotation, while the angular velocity and the moment of inertia with respect to the axis of rotation. The kinetic energy of the rotation, also briefly referred to as the rotational energy can be obtained by
Be expressed. These formulas show the analogy to the corresponding formulas for momentum and kinetic energy of the translational motion.
Examples
Moments of inertia of celestial bodies
Almost all the major body in the universe (stars, planets) are approximated spherical and rotate more or less quickly. The moment of inertia about the rotation axis is always the greatest of the celestial body.
The difference between this ' polar ' and the equatorial moment of inertia is related to the flattening of the body, so its deformation of the pure spherical shape by the centrifugal force of rotation. For Earth, this difference is 0.3 percent, almost equal to the Earth flattening of 1:298,24. In rapidly rotating Jupiter these relative values are about 20 times larger.
Principal moments of inertia of simple geometric bodies
Unless expressly stated otherwise, the focus of the geometric body lies on the rotation axis, referred to by the moment of inertia. is the mass of the rotating body. The moment of inertia for rotations about other axes can then be calculated using the set of Steiner.
Example calculation: Moment of inertia of the homogeneous solid sphere
To calculate the moment of inertia of a solid homogeneous sphere with respect to a rotational axis through the center of the sphere, described in " Calculation" given integral is used. For simplicity, the center of the sphere is to be the origin of a Cartesian coordinate system and extending along the axis of rotation of the axis. To the integral
Evaluate, it is recommended to use instead Cartesian rather spherical coordinates. In this case, the transition to the y Cartesian coordinates x, z and the volume element dV can be expressed by the spherical coordinates. This is done using the substitution rules
And the functional determinants
Provides insertion into the expression for the moment of inertia
This shows the advantage of the spherical coordinates, integral limits do not depend on each other. The two integrations over r and can therefore perform elementary. The remaining integral in
Can be obtained by integration by parts with
Be solved:
For the moment of inertia is finally obtained:
Measurement
To measure a moment of inertia of a body using a rotary table. This consists of a circular disc which is rotatable about its axis of symmetry, and a helical spring ( spiral spring ). It results in a rotation of the disc a rear driving torque is directly proportional to the deflection angle. The constant of proportionality is called the Directorate moment or righting moment. Its value depends on the strength of the spring. The disk is now leading harmonic oscillations with the period of oscillation
From, the moment of inertia of the disk is. If you put an additional body of known moment of inertia of the disk, then the period of oscillation changes to
From the difference of the squares of the respective period of oscillation
Can the Directorate moment the turntable and determine from the above formula, we obtain the moment of inertia of the turntable. If you put any body on the turntable, so you can its moment of inertia with respect to the axis of rotation from the measured oscillation period
Calculate.