Factors Responsible for Moment of Inertia

The moment of inertia, in physics, is the measure of the volume of rotating inertia of a body, i.e., the resistance of the body showing its rotational speed relative to the axis altered by torque.

The axis can be internal or external and can be tilted or straight. The moment of inertia (I) is always specified in relation to that axis. It is defined as the result of adding all the products obtained when each particle in the body is multiplied by the square of its distance from the axis.

The moment of inertia is the name given to rotational inertia, the rotational analogue of the weight of a direct movement. It arises from the virtue of a rotating body to resist a change in its rotational motion. It is also a property due to the density of the material that forms the body. For a given point, the moment of inertia is just a few times the square of a perpendicular distance to the rotating axis, I = mr2.

Examples of the Moment of Inertia

Imagine you are sitting in a moving bus. When it stops after a while, your upper body moves forward, and your lower body does not move. This is because of inertia. Your lower body is connected to the bus, but your upper body is not directly connected to the bus. So when the bus stops, your lower body stops with the bus, but your upper body continues to move forward. That is, it resists changes in its position.

Similarly, your body gets pushed back when you board a moving train. That is because before boarding the train you were relaxed. As soon as you board a moving train, your lower body meets the train, but your upper body is still at rest. Therefore, it is pushed backwards, that is, it is resistant to changes in its position.

The Factors on which the Moment of Inertia Depends

• Mass of the Body 

Mass estimates the amount of matter in a body. It is also the virtue by which a body resists acceleration or a change in its velocity when an external force is applied to it that acts as inertia. The SI base unit of mass is the kilogram (kg). When a body has a higher mass, it is more difficult for an external force to alter its state of motion. Similarly, when a body has a higher mass or when the density of the material with which the body is made is high, its moment of inertia is high. Hence, a higher amount of torque is required to change the body’s rotation.

• Shape and Size of the Body

The moment of inertia is dependent on the rotational axis. The rotational axis of a body depends on the size and shape of the body. As the size and shape alter, the body’s rotational axis alters too. This causes a change in the moment of inertia of the body.

• Axis of Rotation (Distribution of Mass Relative to the Axis)

To measure how the weight of a solid rotating body is distributed relative to the rotating axis, we define a new parameter known as the radius of gyration. It is related to the moment of inertia and the total body weight. When the mass on one side is heavier, the body’s rotational axis is closer to it, and the moment of inertia requires a higher torque on that side to alter the motion.

• Position and Orientation of the Axis of Rotation concerning the Body

It is the placement of the body on the axis and its orientation. If there is a change in the direction of the axis, the direction of the torque will also have to change to bring about a change in the rotation of the body. Similarly, changing the point at which the axis of rotation passes through the body, that is, to change its position, would also change the moment of inertia of the body.

The Formula for the Moment of Inertia

The moment of inertia is given by I = mr2.

Where,

m = Sum of the product of the mass.

r = Distance from the axis of the rotation.

The interim unit of inertia is a composite unit of measurement. According to the international system of units (SI), mass is given in kgs and distance in metres. Hence the dimensions of the moment of inertia are kilogram-metres squared.

The moment of inertia of any body shape that a mathematical formula can define is usually calculated by integral calculus. Moment of inertia of a disk can be measured by cutting it to the number of fixed rings, finding their quantity, multiplying the mass by their distances from the reference point, and combining these products. The merging process is done automatically; the answer is I = (mR2) / 2. 

Rotating bodies can be further categorised as:

• Discrete (System of particles)

• Continuous (Rigid body)

The Moment of Inertia of a System of Particles

For a system of various point particles that rotate about a fixed axis, the moment of inertia can be given as I = ∑jmjrj2, where mj gives the mass of each point particle rj gives the distance of these particles from the axis. Because of the r2 term, the inertia time increases as the square distance to the rotating axis does not change. Moment of inertia is the rotational analogue of weight in linear motion.

The Moment of Inertia of a Rigid Body

When we talk about a rigid body, we assume that all the particles in a rigid body are uniformly distributed. If we were to pick two cross-sections of the rigid body at random such that both the cross-sections are of equal area, then we would have the same number of particles in both the cross-sections. We also assume that the density of the material of the body is fixed and not fluctuating.

Thus we can say that the moment of inertia of a rigid body can be described in terms of the moment of inertia of a system of particles, the only difference in the formula being I=∫r2dm,. The same is true even for oddly shaped rigid bodies.

Conclusion

Moment of Inertia is the name given to rotational inertia, the rotational analogue of the weight of a direct movement. It arises from the relationship of the rotation of the rotating movement. Inlet time should be specified concerning the selected rotation axis. By the point of the point, the moment of inertia is just a few times the square of a perpendicular distance to the rotating axis, I = mr2. That point-of-point relationship becomes the basis of all other inertia times as anything can be built from a set of points.

The moment of inertia is given by I = mr2.

The interim unit of inertia is a composite unit of measurement. According to the international system of units (SI), mass is given in kgs and distance in metres. Hence the dimensions of the moment of inertia are kilogram-metres squared.