Formula for moment of Inertia of different shapes

The term Moment of inertia in physics arises in case of the rotational motion of an object. Moment of inertia has the same mathematical role in rotational motion as that of mass in linear motion. But physical significance of the moment of inertia is different from that of the mass of an object. In this article, we are going to discuss the definition, unit, dimension, physical significance, equation or formula of moment of inertia for different shaped objects like a ring, disk, sphere, rod, etc.

Contents in this article:

  1. Definition of moment of inertia
  2. Mathematical equations of moment of inertia
  3. Units of moment of inertia
  4. Dimension of moment of inertia
  5. Physical significance of moment of inertia
  6. Moment of inertia and mass
  7. Equation for moment of inertia for different objects list

Definition of Moment of inertia

We know that in linear motion, a force produces acceleration in a body. But in rotational motion, a torque can rotate an object and produces angular acceleration in that object. Now, the moment of inertia of a rotating object is defined as the amount of external torque required to produce unit amount of angular acceleration in that object.

Sometimes, the moment of inertia is defined as the multiplication between the mass of the object and the square of the distance of the object from the axis of rotation. But this is not the 100% correct definition. It is just okay.

Moment of inertia is a tensor quantity. But in lower classes, it is considered to be a scalar quantity. The symbol of moment of inertia is I. It depends on the mass of the object and the distance of the object from the axis of rotation. The values of moment of inertia of the same object are different for different axes of rotations.

Equation for Moment of inertia

There are two main mathematical equations for moment of inertia. These are –

\small {\color{Blue} I = mr^{2}} ……….(1)

Where, m is the mass of the object and r is the distance of the object from the axis of rotation.

And, \small {\color{Blue} \tau = I\alpha } ……….(2)

Where, \small {\color{Blue} \tau } is the Torque on the object and \small {\color{Blue} \alpha } is the angular acceleration of the object.

Among the above two formulae of moment of inertia, equation-(1) has most of uses.

Unit of Moment of Inertia

One can easily derive the units of moment of inertia from the equation-(1). The SI unit of moment of inertia is kg.m2 and the CGS unit of moment of inertia is g.cm2.

Dimension of Moment of Inertia

To find the dimensional formula of moment of inertia, we will use the equation-(1) again. The dimension of moment of inertia is [ML2].

Physical Significance of Moment of Inertia

Moment of inertia is the rotational counterpart of the mass. Physically, the moment of inertia represents the amount of external torque is required to produce unit amount of angular acceleration in an object. This gives an idea that how easily an object can be rotated.

Moment of Inertia and Mass

I have already mentioned that the moment of inertia is the rotational counterpart of the mass of an object. Mathematically, the moment of inertia plays same role in rotational motion as the mass plays in linear motion. For example, for linear motion F=ma, where m is the mass of the object, F is the force which is the cause of the linear motion and a is the acceleration in linear motion i.e. the linear acceleration. Now, for rotational motion, \small {\color{Blue} \tau = I\alpha }, where, I is the moment of inertia, \small {\color{Blue} \tau } is the torque which is the cause of rotational motion and \small {\color{Blue} \alpha } is the angular acceleration. So, if we compare each term of both equations then we can see that Moment of inertia has the same role in rotational motion as that of the mass in linear motion.

Formula for Moment of inertia of different shapes

Here I am going to share a chart or list of the moment of inertia formulas for most of the objects like a rod, circle or circular ring, disk, sphere, cylinder, etc. about different axes of rotation. Check the following list of moment of inertia of different objects of different shapes.

Serial No.Shape of the Object Axis of RotationExpression for Moment of Inertia
1Uniform RodAxis is perpendicular to the length of the rod and passing through one of its end\small I=\frac{1}{3}ML^{2}
2Uniform Rod Axis is perpendicular to the length of the rod and passing through the center of mass of the rod\small I=\frac{1}{12}ML^{2}
3Circle or circular ringAxis is perpendicular to the plane and passing through the center of mass of the ring\small I=MR^{2}
4Circle or circular ring About the diameter of the ring\small I=\frac{1}{2}MR^{2}
5Uniform DiskAxis is perpendicular to the plane and passing through the center of mass of the disk\small I=\frac{1}{2}MR^{2}
6Uniform Disk About the diameter of the ring \small I=\frac{1}{4}MR^{2}
7Hollow SphereAbout its diameter \small I=\frac{2}{3}MR^{2}
8Solid SphereAbout its diameter \small I=\frac{2}{5}MR^{2}
9Hollow CylinderAbout its own axis\small I=MR^{2}
10Hollow CylinderAbout the axis perpendicular to its length and passing through the center of mass\small I=\frac{1}{12}ML^{2} + \frac{1}{2}MR^{2}
11Solid CylinderAbout its own axis \small I=\frac{1}{2}MR^{2}
12Solid CylinderAbout the axis perpendicular to its length and passing through the center of mass \small I=\frac{1}{12}ML^{2} + \frac{1}{4}MR^{2}
List of moment of inertia of different shapes

Here, L is the length of the rod and cylinder, R is the radius of the circle or ring, disk, sphere and cylinder and M represents the mass of each object. These parameters L, R and M do not have the equal values in each body. The value varies body to body. Here, I have just taken the same symbols to write mathematical expressions only.

Some important questions

  1. On which parameters the moment of inertia depends? Answer: Moment of inertia depends on the mass of the object, Axis of rotation of the object and the distance of the object from the axis of rotation.
  2. Is moment of inertia a scalar quantity or vector quantity? Answer: Moment of inertia is a Tensor quantity. But in junior classes it is considered as a scalar quantity.

This is all from the definition, unit, dimension, physical significance and formula for moment of inertia of different objects. If you have any doubt on this topic you can ask me in the comment section.

Thank you!

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