A Simple Note on Moment of Inertia

Before Galileo Galilei found the concept of the moment of inertia, scientists believed that an external force is required to move a body, keep it in motion, and stop a body. They thought that a body stops as soon as the force applied to it is withdrawn.

For example, if we are rowing a boat, it keeps moving in water, but the boat stops moving forward as soon as we stop rowing. These observations supported that force was required to keep a body moving. This concept was objected to by Galileo, who found out through his experiments that a body can keep moving even without applying an external force on it.

Inertia

According to the inertia, or Galileo’s law of inertia, a body in a state of rest continues to be in a state of rest, and a body in a state of motion continues to be so until an external force is applied to it. This law of inertia was later used by Newton, who said, “If a body is in a state of motion, it keeps moving in the same direction, and if it is in a state of rest, it remains in a state of rest until an external force is applied on it.” This is known as Newton’s first law of motion.

Inertia is a property of a body by which it resists any force applied to it. For example, if a mobile phone is kept somewhere on a tabletop, it will remain there until someone else picks it up or displaces it.

Mass and Inertia

Inertia is an inherent property of the body which depends on its mass. Inertia is directly proportional to the body’s mass, which means the more the mass constituted in a body, the more is its inertia and vice-versa. Some examples of inertia can be witnessed in our day-to-day life:

• When a bus having passengers suddenly starts moving, passengers experience a backward force
• A bowler takes a long run before throwing the ball to the batsman.

Moment of Inertia

An object’s moment of inertia is a calculated value for a stiff body revolving around a fixed axis. The axis may be internal or external, and it may or may not be fixed. However, the moment of inertia (I) is always expressed in terms of that axis.

The moment of inertia is determined by the spreading of mass around the rotational axis. The MOI varies based on which axis is selected. That is, the same item may have different values related to the moment of inertia depends on the place and direction of the rotation axis.

The moment of inertia is also known as the angular mass or rotational inertia.

The SI unit of moment of inertia is kg m².

The moment of inertia depends on the following factors:

1. Mass of the body
2. Size as well as the shape of the body
3. Distribution of mass along the axis of rotation of the body

These factors are true for a body in translation motion. If a body is in rotational motion, the moment of inertia starts depending on its angular velocity.

Moment of Inertia of a System of Particles

The summation of all the masses and square of the distance from the axis of rotation for translational motion can be written as:

I = Σmr²

Where m = sum of all the masses present in a system

r = sum of distances from the point of rotation

For rotational motion moment of inertia is expressed as:

I = 2K/w²

Where K = Kinetic energy of the body

w = angular velocity of the body

The moment of inertia is expressed in Kg m² because it is a derived quantity that depends on mass and square of the distance.

Conclusion

The moment of inertia is thus an inherent property of a body that helps the body resist any external torque applied to it. It is calculated using integration and has various theorems involved. The moment of inertia does have a direction and depends on the mass, shape, and distribution of mass of an object.; thus, it is a tensor quantity that acts as both scalar and vector quantity depending on the situation. It is not called a vector quantity because it does not follow vector algebra.

Various examples can be illustrated to understand the moment of inertia. Its discovery by Galileo Galilei gave Newton a better understanding of things. He further used inertia to understand motion, and the inertia was redefined and used by him as his first law of motion.