What are products of inertia?
What are products of inertia?
The products of inertia are the measure of symmetry in a body. Products of inertia can be calculated for mass, area, and volume. If one plane is symmetric then, the product of inertia for the two planes perpendicular to that plane is zero. For example, if the x-y plane is symmetric, then products of inertia about x-z and y-z planes are zero. In cases where the body is symmetric about 2 axes, the product of inertia is zero about each of the three planes. To calculate the product of inertia in a rigid body, a small element of a rigid body is considered and then the product of inertia of that small element is calculated. The result is then integrated for the entire body to obtain the product of inertia of the rigid body.
The product of inertia of mass of a rigid body about the x-y plane is calculated by the formula,
Here, 
is the product of inertia of mass of the rigid body about the x-y plane, m is the mass of the rigid body, and dm is the mass of the small element of the rigid body.
The product of inertia of mass of a rigid body about the y-z plane is calculated by the formula,
Here, 
is the product of inertia of mass of the rigid body about the y-z plane.
The product of inertia of mass of a rigid body about the x-z plane is calculated by the formula,
Here, 
is the product of inertia of mass of the rigid body about the x-z plane.
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