Parallel Perpendicular Axes Theorem

The parallel axis theorem is used to estimate the moment of inertia of a rigid body around an axis that is parallel to the object line’s mass (about which the moment of inertia is known). The perpendicular axis theorem notes that the moment of inertia of a rigid body along an axis that is at right angles to two other axes about which its moment of inertia is known is equal to the sum of the rigid body’s moments of inertia about the two known axes. The parallel axis theorem is also known as the Huygens–Steiner theorem, after Christiaan Huygens and Jacob Steiner.

Moment of Inertia

We also know that inertia is a quality that a body possesses in order to resist changes in its linear state of motion or rest. This is a property that determines the body’s mass. The moment of inertia, indicated by the symbol (I), is just a measure of a body’s ability to resist its condition of rotational motion. It plays the same exact role in rotational motion that mass does in linear motion, therefore the two can be compared. The parallel axis theorem and the perpendicular axis theorem are two theorems on moment of inertia. As a result, we can describe a moment of inertia as a body’s ability to resist any change in its condition of uniform motion or rest.I=Mr2, where r is the particle’s perpendicular distance from the axis of rotation and M is the mass of the rotating body.

NOTE- 1. The moment of inertia is also a scalar and no vector quantity. It’s a tensor quantity, actually.

2. This is not a constant for a body because it is dependent on the rotation axis.
3. The greater a body’s mass, the greater its moment of inertia.
4. The greater the moment of inertia, the more the mass is divided from the axis.

Parallel axis Theorem

The Huygens–Steiner theorem, called after Christiaan Huygens and Jacob Steiner, is another name for the parallel axis theorem. 

According to the parallel axis theorem, the moment of inertia of a body about an axis parallel to the body passing through its core is equal to the sum of the moment of inertia of a body about the axis passing through the center the product of the mass of the body times the square of the distance between the two axes.

Parallel Axis Theorem Formula

Parallel axis theorem statement is as follows:

I=Ic+Mh2

Here,

 I is the body’s moment of inertia.

 Ic denotes the moment of inertia around the centre.

 m is the body’s mass

h2  distance between the two axis squared

Example for Parallel Axis Theorem

Question-   If the moment of inertia of a body along a perpendicular axis passing through its centre of gravity is 50 kg∙m2 and the mass of the body is 30kg. What is the moment of inertia of that body along another axis which is 50cm away from the current axis and parallel to it? Use Parallel Axis Theorem Formula

Sol: By using Parallel axis theorem,

I=IG+Mb²

I=50+(30×0.5²)

I=57.5kg-m²

Perpendicular Axis Theorem

A perpendicular axis theorem states that the f inertia of a planar lamina (i.e. 2-D body) about an axis perpendicular to the plane of the lamina is equal to the sum of the moments of inertia of the lamina about two axes at right angles to each other, intersecting each other at the point where the perpendicular axis passes through it. 

Define perpendicular axes x,y and z (which meet at origin  O) so that the body lies in the xy plane, and the z axis is perpendicular to the plane of the body. Let IX,IY and IZ be the moment of inertia about axis X, Y , Z respectively. Then the perpendicular axis theorem states that

Iz=Ix+Iy

If a planar object has rotational symmetry such that Ix and Iy  are equal, then the perpendicular axes theorem provides the useful relationship:

Iz=2Ix=2Iy

Let us seen an example of this theorem:

Let’s suppose we want to find the moment of inertia of a uniform ring as a function of its diameter. Let MR2/2 be its middle, where M denotes mass and R denotes radius. As a result of the perpendicular axes theorem, Iz=Ix+Iy. All of the diameters are equal since the ring is uniform.

∴Ix=Iy

∴Iz=2Ix

Iz=MR²/4

At last the moment of inertia of a disc about any of its diameter is MR²/4

Applications of Parallel and perpendicular axis theorems

A moment of inertia of a rigid body around any axis is calculated using the parallel and perpendicular axis theorems in combination. The moment of inertia of any rotating object can be calculated using the parallel and perpendicular axis theorems.

Conclusion

We know that inertia is a quality that a body possesses in order to resist changes in its linear state of motion or rest. This is a property that determines the body’s mass.

. The moment of inertia is neither a scalar and nor vector quantity. It’s a tensor quantity, actually.

. The greater a body’s mass, the greater its moment of inertia.

The Huygens–Steiner theorem, called after Christiaan Huygens and Jacob Steiner, is another name for the parallel axis theorem. 

Parallel axis theorem formula: I=Ic+Mh²

Perpendicular axis theorem formula : Iz=Ix+Iy