Moment of Inertia of Rigid Bodies

Inertia is a characteristic of a body that opposes any force that seeks to move it or, if it is moving, to modify the amount or direction of its velocity. Inertia of a rigid body is a passive attribute that allows a body to do nothing except oppose active agents such as forces and torques. A moving body continues to move not because of inertia, but because there is no force to slow it down, modify its trajectory, or accelerate it up.

Moment of Inertia of rigid bodies

The moment of inertia of rigid bodies determines how difficult it is to rotate an exacting body along a certain axis. Newton’s first law of motion states that unless compelled by an external source known as force, a body must continue in its condition of rest or uniform motion. Inertia is defined as a material body’s incapacity to change its state of rest or uniform motion on its own.

The basic attribute of matter is inertia. The moment of inertia of a rigid body is the mass attribute that influences the torque required to achieve a given angular acceleration about an axis of rotation. It is a natural attribute of stuff. A body resists any change in its condition of rest or uniform motion along a straight line due to inertia. For a given force, the bigger the mass, the greater the resistance to motion or the greater the inertia. In translatory motion, the mass of the body is used to calculate the coefficient of inertia.

Similarly, in rotational motion, a body that is free to spin along a particular axis resists any change in its condition that is desirable. The amount of opposition will be determined by the mass of the body and its distribution along the axis of rotation. The moment of inertia of the rigid body about the specified axis is the coefficient of inertia in rotational motion.

Significance

The significance of moment of inertia of a rigid body are as follows:

•   The greater the concentration of mass away from the axis, the greater the moment of inertia. The moment of inertia of a rigid body varies with respect to the different axes of rotation.
•   It is significant in rotatory motion. It is the property of the body that opposes state change in rotatory motion.
•   It is connected to rotational motion in the same way that mass is related to translational motion.

Unit of Moment of Inertia

Having defined the Moment of Inertia of a rigid body, we can now know its unit.  

The unit of moment of inertia of a rigid body is a composite unit of measurement. In the International System (SI), m is measured in kilograms and r is measured in meters, with I (moment of inertia) having the dimension kilogram-meter square.

Calculation of Moment of Inertia of Rigid body

Having known the factors on which the moment of inertia of rigid bodies depends on, we can now calculate the moment of inertia of rigid bodies.

When an item has a continuous distribution of mass, the moment of inertia of a rigid body may be computed by integrating the moments of inertia of its component pieces. If dm is the mass of any infinitesimal particle of the body and r is its perpendicular distance from the axis of rotation, the moment of inertia of the rigid body about the axis is given as:

I= ∫r²dm

Factors affecting moment of inertia

The moment of inertia of a rigid body depends on the following factors:

•   Body mass index.
•   Body dimensions and form
•   Mass distribution about the axis of rotation
•   The position and direction of the axis of rotation in relation to the body.

Moment of Inertia of Different Objects:

We know that the moment of inertia of a rigid body depends on various factors.

Hence, it is different for different rigid bodies. The moment of inertia of different rigid objects are as follows:

•   Rod–       Through center: 1 ⁄ 12 Ml²

–        Through end:  1 ⁄ 3Ml²

•   Sphere

–        Solid sphere:  2 ⁄ 5mr²

–        Hollow sphere:  2 ⁄ 3mr²

• Hoop

–        Hoop about diameter:  1 ⁄ 2MR²

–          Hoop about symmetrical axis: MR²

•   Cylinder

–        Solid cylinder or disc, symmetrical axis:  1 ⁄2MR²

–        Solid cylinder, central diameter:  1 ⁄ 4MR² +  1 ⁄ 12ML²

•   Rectangular objects

–        Solid rectangular box: 1 ⁄ 12m (h² + w²)

–        Solid rectangular plate:  1 ⁄ 12 m (h² + w²)

• Ring: mr²
• Disc:  1 ⁄ 2mr²

Conclusion

The axis of rotation influences the moment of inertia. After picking two unique axes, the item resists rotational change in different ways. We establish a new parameter known as the radius of gyration to describe how the mass of a rotating rigid body is distributed with respect to the axis of rotation. It is connected to the moment of inertia and total body mass. It’s worth noting that we can write I = Mk2 where k is the length dimension.

As a result, the radius of gyration is the distance from the axis of a mass point whose mass equals the mass of the entire body and whose moment of inertia equals the body’s moment of inertia about the axis. As a result, the moment of inertia is affected not only by the mass, form, and size of the body, but also by the distribution of mass inside the body about the axis of rotation.