# Statement of Parallel Axis Theorem: Notes pdf ppt Engineering Mechanics

## Parallel Axis Theorem:

The theorem determines the moment of inertia of a rigid body about any given axis, given that moment of inertia about the parallel axis through the center of mass of an object and the perpendicular distance between the axes.

## Statement:

The moment of inertia about Z-axis can be represented as: Where
Icmis the moment of inertia of an object about its centre of mass
m is the mass of an object
r is the perpendicular distance between the two axes.

## Proof:

Assume that the perpendicular distance between the axes lies along the x-axis and the centre of mass lies at the origin. The moment of inertia relative to z-axis that passes through the centre of mass, is represented as Moment of inertia relative to the new axis with its perpendicular distance r along the x-axis, is represented as: We get, The first term is Icm,the second term is mr2and the final term is zero as the origin lies at the centre of mass. Finally,

## For Area Moments of Inertia:  : is the cross-sectional area. : is the perpendicuar distance between the centroidal axis and the parallel axis.

## For Area Radius of Gyration:  : is the Radius of Gyration about an axis Parallel to the Centroidal axis. : is the Radius of Gyration about the Centroidal axis. : is the perpendicuar distance between the centroidal axis and the parallel axis.

## For Mass Moments of Inertia:  : is the mass of the body. : is the perpendicuar distance between the centroidal axis and the parallel axis.

## For Mass Radius of Gyration:  : is the Radius of Gyration about an axis Parallel to the Centroidal axis. : is the Radius of Gyration about the Centroidal axis. : is the perpendicuar distance between the centroidal axis and the parallel axis.

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