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# MomentOfInertia

MomentOfInertia[reg,pt,v]

computes the moment of inertia for the region reg rotating around an axis through the point pt in direction v.

MomentOfInertia[reg]

computes the moment of inertia matrix for the region reg relative to the center of mass.

MomentOfInertia[reg,pt]

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 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.  »

# Examples

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## Basic Examples(4)

The moment of inertia when rotating around the axis through and in direction :

 In[1]:=
 Out[1]=
 In[2]:=
 Out[2]=

If the vector is omitted, you get a moment of inertia matrix about that point:

 In[1]:=
 Out[1]=

Using this matrix and a normalized vector, the moment of inertia around any axis can be found:

 In[2]:=
 Out[2]=

If the point is omitted, you get a moment of inertia matrix around the center of mass:

 In[1]:=
 Out[1]=

This is equivalent to specifying the center point by RegionCentroid:

 In[2]:=
 Out[2]=

Compute the moment of inertia for a region with symbolic parameters:

 In[1]:=
 Out[1]=