**Branch :**First Year-Engineering Syllabus

**Subject :**Elements of Mechanical Engineering

## Translation of Coordinate Axes

** Translation of Coordinate Axes: **The defining equations for the moments and products of inertia, as given by

and , ..eq..0

do not require that the origin of the Cartesian coordinate system be taken at the mass center. Next, one can calculate the moments and products of inertia for a given body with respect to a set of parallel axes that do not pass through the mass center. Consider the body shown in Fig. (given below). The mass center is located at the origin O, ≡ C of the primed system x'y'z'. The coordinate of O' with respect to the unprimed system xyz is (x_{c},y_{c}, z_{c}). An infinitesimal volume element dV is located at (x,y, z) in the unprimed system and at (x',y', z') in the primed system. These coordinates are related by the equations

________________eq..1

Themoment of inertia about the x-axis can be written in terms of primed coordinates by using

and

______________________eq..2

where m is the total mass of the rigid body, and the origin of the primed coordinate system was chosen at the mass center. One can write

_______________eq..3

and therefore, the two integrals on the right-hand side of are zero. In a similar way, one can obtain Iyy and I. The results are summarized as follows:

__________________eq..4

or, in general, ___________________eq..5

where d is the distance between a given unprimed axis and a parallel primed axis passing through the mass center C. Equation (5) represents the parallel−axes

fig....(2)Rotation of coordinate axes

Theorem. The products of inertia are obtained in a similar manner, using (0) and (1)

The two integrals on the previous equation are zero. The other products of inertia can be calculated in a similar manner, and the results can be written as follows:

eq...6 Equations (4) and (6) shows that a translation of axes away from the mass center results in an increase in the moments of inertia. The products of inertia may increase or decrease, depending upon the particular case.