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State and explain Parallel Axis Theorem

Referencing Styles : APA | Pages : 1

The parallel axis theorem which is also known as the Steiner’s theorem which is named after Christian Huygens and Jacob Steiner. This theorem is basically used for the determination of the mass moment regarding inertia or the moment of area that is second regarding a rigid body about any of the axis, given the body’s moment of inertia about a parallel axis through the object’s centre of gravity as well as the perpendicular distance between the axes.

This theorem quantifies the variation in regards to the moment of inertia that is in relation with the distance of the regarding the distance of the rotation of axis form the mass centre. When the rotation axis of a body passes through the centre of mass, the moment of inertia is depicted to be minimum, whereas the rotation axis of a body passes through the centre of mass, the moment of inertia is depicted to be maximum.

The application regarding the parallel axis theorem for the rotation axis is depicted to be offset from the mass centre that is presented in the figure below:

 

In accordance to the stated parallel axis theorem, the moment of inertia is said to be the sum of the moment of inertia via the centre of mass as well as the mass product and the square of the perpendicular distance in between the centre of mass as well as rotation of axis.

 

In this section the moment of inertia at the centre of mass is Icm, where mass is m as well as the perpendicular distance in between the axis of rotation and the centre of mass is depicted as d.

Apart from mass of the moment of inertia, the parallel axis theorem is depicted to be utilized for the calculation of the area moment of inertia (IArea) as well as radius of gyration (k).

Area moment of inertia,

 

Radius of gyration,

The parallel axis theorem is applicable to the bodies of any shape. This theorem states that the moment of inertia regarding a body about an axis that is parallel to an axis that passes through the centre of mass is equal to the sum regarding the moment of inertia of a body that passes through the axis of the centre of mass as well as the product of the mass and the square of the distance present within the axes.

IZ’ = I+ Mα²

Where, α is the distance between two axes.

This theorem may be explained with the help of some easy examples that are stated below:

The moment of inertia regarding a thin and uniform rod that possess mass M and length L about an axis that is perpendicular to the rod and through its centre is I. The moment of inertia regarding the rod about an axis perpendicular to the rod through its endpoint is:

Icentre = M L² / 12 and Iendpoint = M L² / 3 = 4 I

Thus with the help of the above example the parallel axis theorem is well explained.

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