Statics/Moment of Inertia (contents)

The moment of inertia can be defined as the second moment about an axis and is usually designated the symbol I. The moment of inertia is very useful in solving a number of problems in mechanics. For example, the moment of inertia can be used to calculate angular momentum, and angular energy. Moment of inertia is also important in beam design.

Shape moment of inertia

The area moment of inertia takes only shape into account, not mass.

It can be used to calculate the moment of inertia of a shape about the x or y axis when I is only important at one cross-section.

$I_x=\int y^{2}dA$

$I_y=\int x^{2}dA$

$I_z=\int r^{2}dA$

Because $r=x^2+y^2$ the following is true $I_z=I_x+I_y$

Mass moment of inertia

The mass moment of inertia takes mass into account. The mass moment of inertia of a point mass about a reference point is equal to mass multiplied by the square of the distance. The metric units are kg*m^2.

$I_m=\int r^{2}dm$

Radius of gyration

The radius of gyration is the radius at which you could concentrate the entire mass to make the moment of inertia equal to the actual moment of inertia. If the mass of an object was 2kg, and the moment of inertia was $18kg*m^2$, then the radius of gyration would be 3m. In other words, if all of the mass was concentrated at a distance of 2m from the axis, then the moment of inertia would still be $18kg*m^2$. Radius of gyration is represented with a k.

$k=\sqrt{I/m}$

The formula for the area radius of gyration replaces the mass with area.

$k=\sqrt{I/A}$

Parallel axis theorem

If the moment of inertia is known about one axis, and the object will spin about a different axis a given distance away; then, the moment of inertia can be found by the following formula,

$I_{newaxis}=I_{knownaxis}+(mass)d^2$

where d is the distance between the two axis of rotation and the axis of known I.