computes the moment of inertia for the region reg rotating around an axis through the point pt in direction v.
computes the moment of inertia matrix for the region reg relative to the center of mass.
computes the moment of inertia matrix relative to the point pt.
Details and Options
- Moment of inertia is also known as rotational inertia, area moment of inertia, and mass moment of inertia. The moment of inertia matrix is also known as rotational inertia matrix and angular mass matrix.
- Moment of inertia is the resistance to rotational acceleration for rigid bodies and is the rotational analog of mass, which is the resistance to translational acceleration.
translational acceleration force , acceleration , mass rotational acceleration torque , rotational acceleration , moment of inertia
- MomentOfInertia[reg,pt] gives the moment of inertia matrix about the point pt and is given by
2D moment of inertia matrix 3D moment of inertia matrix
- where reg is the region reg translated by -pt.
- The moment of inertia matrix ℐ can be used to compute the moment of inertia for any direction v through the formula , where in 2D it is required that v be in the - plane. »
- MomentOfInertia computes a result under the assumption that the mass density of the region is constant.
- For varying mass density , use Integrate or NIntegrate to compute the corresponding moment of inertia matrix according to the following formula. »
Examplesopen allclose all
Basic Examples (4)
This is equivalent to specifying the center point by RegionCentroid:
Special Regions (15)
Rectangle with length and width :
Disk with radius :
Annulus with radii and :
StadiumShape with length and radius :
Right Triangle with sides and :
Cuboid with length , width , and height :
Ball with radius :
Ellipsoid with semiaxes , , and :
SphericalShell with radii and :
Cylinder with length and radius :
CapsuleShape with length and radius :
Rectangular Tetrahedron with sides , , and :
Rectangular Pyramid with sides , and height :
Rectangular Prism with sides , and height :
Formula Regions (2)
Mesh Regions (2)
Derived Regions (3)
The mass of a body with a constant mass density is given by m=ρ Volume[body]. Express the moment of inertia in terms of mass of the body:
The only other nonzero entry of the matrix is on the diagonal. It is the polar moment of area about the axis and, by the perpendicular axis theorem, it is equal to the sum of the other two entries on the diagonal:
A moment of inertia matrix can be thought as a matrix defining an Ellipsoid:
Use Eigensystem to find the principal axes:
The angular momentum for a rotating rigid body is given by , where is the moment of inertia and is the angular velocity. Suppose a ball of radius 1 and uniform density 1 rotates around the axis on a string with adjustable length . When , the angular velocity is . Find the angular velocity as a function of :
The kinetic energy for a rotating rigid body is given by , where is the moment of inertia and is the angular velocity. Compute the kinetic energy of a ball with radius and uniform density , rotating with angular velocity around an axis passing through its center:
A kinetic energy recovery system (KERS) stores kinetic energy for a car when braking. Suppose the energy is stored in a rotating steel cylinder with radius and length . How fast does the cylinder need to rotate to store of energy, assuming the density of steel is :
The kinetic energy for a rigid body is given by , where is an angular velocity vector and is the moment of inertia matrix. Find the rotational energy for a cuboid with sides 2, 4, and 6 when rotating around different axes. In all cases, the mass of the rotating body is the same, but how the mass is distributed relative to the rotation axis differs and is represented by moment of inertia:
For a rotating rigid body, the relation between the torque and the angular acceleration is given by , where is the moment of inertia. Compute the torque required to give a cube with one-meter sides of lead and angular acceleration of :
The parallel axis theorem gives a relation between the moment of inertia around an arbitrary axis and the moment of inertia around the parallel axis passing through the center of mass, where is the total mass and is the distance between the axes:
Properties & Relations (11)
If the point is omitted, the RegionCentroid is taken as a default:
Use Integrate to compute moment of inertia for a body varying mass density :
Moment of inertia matrix in 2D can be computed using RegionMoment:
Moment of inertia matrix in 3D can be computed using RegionMoment:
The moment of inertia of a Point is the square of the distance of the point to the axis: