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:

Where
I_cmis 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 I_cm,the second term is mr²and the final term is zero as the origin lies at the centre of mass. Finally,

 Parallel Axis Theorem: Transfer of Axis Theorem

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.

Feedback is important to us.