Moment Of Inertia Of Different Shapes And Objects

In science, the phrase “moment of inertia or the MOI” refers to an object’s rotary movement. In rotary movement, the MOI plays the very same quantitative role that mass does in linear movement. However, the physical meaning of the MOI differs from the very same meaning of an object’s mass. The description, measurement, quantity, physical importance, formula or equation of the angular momentum for various shaped objects such as a circle, cylinder, spheres, rod, and so on are different from each other. We’ll see the same here, i.e., the moment of inertia for different shapes and objects and also the moment of inertia formulas for different shapes and objects.

But before that let us understand in a small para what a moment of inertia is and why it is different for different shapes and objects

Moment of Inertia

The MOI can be said as a tensor quantity. However, it is regarded as a scalar in the lower orders. The Moment of Inertia is referred to with the symbol ‘I’. It is determined by the item or object’s mass and its displacement from the rotational axis. For various rotational axis, the MOI has varying values for the very same item. The Rotational Inertia of the Moment of Rotation is also referred to as the Moment of Inertia. The equation for finding the moment of inertia is as,
I=m × r²

Where,

I = Moment of inertia

m = object’s mass

r = object’s distance from the rotational axis

And every shape has its different MOI because its weight or mass distribution is different from others. And also the axis of rotation has its importance. It can be of two kinds, through the object or from the outside of the object.

Let us discuss the MOI formulas for different shapes and objects

MOI of a rectangular plate

The Moment of Inertia of a rectangular plate has a length of ‘l’ and width of ‘b’ and the centre of the axis is passing through the centre of mass of the plate and is perpendicular to the plate. 

I=1 ⁄ 12M(l²+b²)

MOI of a uniform rod

When the axis of rotation is perpendicular to the particular rod and it passes through one of the ends, the MOI, in that case, would be,

I=1 ⁄ 3ML²

And when the axis of rotation is the same as above and is passing through the centre of mass of the rod. The MOI would be,

I=1 ⁄ 12ML²

MOI of a Circle/Ring

MOI is when the axis of rotation of the particular circle of the ring is passing through the centre and is perpendicular to the plane of the circle. 

I=MR²

And when the axis of rotation is about the diameter of the particular circle the moment of inertia would be,

I=1 ⁄ 2MR²

MOI of a Hollow Sphere

The moment of inertia of a hollow sphere when the axis of rotation is passing through the centre or about the diameter of the hollow sphere is, 

I=2 ⁄ 3MR²

MOI of a Solid Sphere

The moment of inertia of a solid sphere when the axis of rotation is passing through the centre or about the diameter of the hollow sphere is, 

I=2 ⁄ 5MR²

MOI of a hollow Cylinder

MOI or the moment of inertia of a hollow cylinder when the axis of rotation is passing about the axis of the particular hollow cylinder is,

I=MR²

And when the axis of rotation is passing through the centre of mass of the cylinder and is perpendicular to the length of the cylinder will be,

I=1 ⁄ 12ML²+1 ⁄ 2MR²

MOI of a solid cylinder

The moment of inertia of a solid cylinder has quite a similar axis of rotations as of a hollow cylinder. The equation if it when the rotational axis is passing through the cylinder’s axis is,

I=1 ⁄2MR²

And when through the centre of mass and perpendicular to the length it will be,

I=1 ⁄ 12ML²+1 ⁄ 4MR²

MOI of a Disk

MOI when the disk’s axis of rotation is passing through its centre and perpendicular to the place,

I=1 ⁄ 2MR²

And when through the diameter of the disc it will be,

I=1 ⁄ 4MR²

In all the above equations,

I = Moment of Inertia

R = radius of the cylinder/sphere

M = Mass of each particular object or particle

Conclusion

To sum up, The moment of inertia is the resisting force for any angular momentum or the torque imposed on any object. The formula for the moment of inertia is I=m ×r². This is a general equation. The moment of inertia of different shapes and objects as well as the moment of inertia formula of different shapes and objects does not remain the same. For a rectangular plate, it is different, for a sphere it is different and for a rod it is different. Also, it changes depending on the axis of rotation about which the object is rotating. The equations for different shapes and objects are given above in this article.