Rotational Inertia Formula

The rotational inertia can be defined as an analogous mass in the case of linear motion. It describes the relationship under the dynamics of the rotational motion. The moment of inertia needs to be specified with respect to some axis chosen for the rotation. It depends on the shape of the object as well as on the axis of rotation. The rotational inertia differs for different objects and varies according to their axis of rotation. Rotational inertia is important in many problems of physics which involve mass in rotational motion. In this topic, we will discuss the concept and Rotational Inertia Formula with examples. Let us learn the interesting concept in an easy way!

                                                                                                                                           Source:  en.wikipedia.org

Rotational Inertia Formula

What is the rotational inertia?

Rotational inertia is a property of any object which can rotate along some axis. It is a scalar value and it tells us how difficult it is to change the rotational velocity of the object around some given rotational axis.

Rotational inertia plays a similar role in rotational mechanics as the mass in linear mechanics. Actually, the rotational inertia of an object depends on its mass. Also, it depends on the distribution of that mass along the axis of rotation.

It is used to calculate angular momentum and allows us to explain how rotational motion changes when the distribution of mass changes. It is needed to find the energy which is stored as rotational kinetic energy in a spinning flywheel.

When a mass moves further from the axis of rotation, then it becomes increasingly more difficult to change the rotational velocity of the system. This is because the mass is now carrying more momentum with it around the circle of motion.

It is also due to the momentum vector which is changing more quickly. Both of these will affect and it depends on the distance from the axis.

Formula for Rotational Inertia

Rotational inertia is given the symbol I. For a single body like the tennis ball of mass m. It is rotating along a circle with a radius r from the axis of rotation. Then the rotational inertia will be:

I = m r²

Where,

I	Rotational Inertia
m	Mass of object
r	The radius of the rotational path

Consequently, the rotational inertia has SI units of kg m².

Rotational inertia is also referred to as the moment of inertia. It is also sometimes called the second moment of the mass. Here, the ‘second’ refers to the fact that it depends on the length of the moment arm squared.

Solved Examples

Q.1: Determine the rotational inertia if a 20 kg object is rotating around a circular path of radius 7m?

Solution:

Given:

m = 20 kg

r = 7 m

Rotational inertia Formula is given by,

I = mr²

= 20 ×times 7²

= 980 kg m²

Therefore the rotational inertia of the object will be 980 kg m²

Q.2: An object of mass 3kg is rotating a circular path. It is having rotational inertia of 300 Kg m². Compute the radius of the circular path.

Solution:

Given:

m = 3kg

I = 300 kg m²

Rotational inertia Formula is:

I = mr²

Rearranging the formula to compute the radius of the path,

r = \(\sqrt{\frac{I}{m}} \)

r = \(\sqrt{\frac{300}{3}} \)

2 = \(\sqrt{100} \)

\(r = 10 \;m.\)

Therefore the radius of the circular path is 10 m.