Define moment of inertia (M.I.).

Short Answer:

The moment of inertia (M.I.) is a property of a body that measures its resistance to rotation about a given axis. It depends on both the mass of the body and how that mass is distributed relative to the axis of rotation. In other words, it is the rotational equivalent of mass in linear motion. The greater the moment of inertia, the harder it is to change the body’s angular motion. Mathematically, the moment of inertia is expressed as , where is the perpendicular distance of each small mass element from the axis of rotation.

Detailed Explanation :

Moment of Inertia (M.I.)

In Engineering Mechanics, the moment of inertia (M.I.) is an important concept that describes how mass is distributed with respect to an axis and how it affects the rotational motion of a body. It is sometimes called the rotational inertia because it represents the body’s resistance to angular acceleration, just as mass represents resistance to linear acceleration.

The moment of inertia is a scalar quantity but always associated with a specific axis of rotation. It plays a major role in the analysis of rotating systems such as flywheels, gears, shafts, pulleys, and rotors.

Definition

The moment of inertia can be defined as:
“The moment of inertia of a body about a given axis is the sum of the products of the mass elements of the body and the squares of their perpendicular distances from that axis.”

Mathematically,

where,

  • = Moment of inertia,
  • = Perpendicular distance of each mass element from the axis of rotation,
  • = Small mass element of the body.

If the body is made of uniform density , and volume , the mass element can be written as . Hence,

Thus, the moment of inertia depends on:

  1. The mass of the body, and
  2. The distribution of mass about the axis.

Physical Meaning

The moment of inertia expresses how difficult it is to change the rotational motion of a body.

  • If the mass is concentrated closer to the axis, the M.I. is smaller.
  • If the mass is distributed farther from the axis, the M.I. is larger.

It is analogous to mass in linear motion, where Newton’s second law for rotation is written as:

where,

  • = Torque applied,
  • = Angular acceleration,
  • = Moment of inertia.

This shows that for the same torque, a body with a larger moment of inertia will experience less angular acceleration.

Units of Moment of Inertia

The moment of inertia is expressed as:

Hence, its unit in SI system is kg·m², and in CGS system it is g·cm².

Mathematical Formulas for Moment of Inertia of Simple Bodies

  1. Thin Rod about its center:

about one end:

  1. Rectangular Plate about an axis through its center and parallel to its base:
  1. Solid Cylinder about its central axis:
  1. Hollow Cylinder about its central axis:
  1. Solid Sphere about a diameter:
  1. Hollow Sphere about a diameter:
  1. Circular Disc about an axis through its center and perpendicular to its plane:

These results are widely used in mechanical design and structural engineering applications.

Radius of Gyration

The radius of gyration (k) is another useful term related to the moment of inertia. It represents the distance from the axis of rotation at which the entire mass of the body can be assumed to be concentrated without changing the moment of inertia.

It is defined as:

or,

The radius of gyration gives an intuitive idea of how far the mass is spread from the axis.

Parallel Axis Theorem

The Parallel Axis Theorem is used to find the M.I. of a body about any axis parallel to an axis through its centroid.

It states that:

where,

  • = Moment of inertia about the centroidal axis,
  • = Perpendicular distance between the two axes,
  • = Mass of the body.

This theorem is very useful for composite shapes or when the rotation axis does not pass through the centroid.

Perpendicular Axis Theorem

This theorem applies only to plane (2D) bodies. It states that:

where,

  • = Moments of inertia about the two perpendicular axes in the plane,
  • = Moment of inertia about the axis perpendicular to the plane.

Applications of Moment of Inertia

  1. Rotational Motion Analysis:
    Helps in determining angular acceleration and kinetic energy of rotating bodies.
  2. Design of Machine Components:
    Used in designing flywheels, shafts, pulleys, and rotors for stability and efficiency.
  3. Structural Engineering:
    Helps in finding bending stresses and deflections in beams and columns.
  4. Automobile and Aerospace Design:
    Used to balance rotating parts like engines, propellers, and gyroscopes.
  5. Energy Storage Systems:
    Flywheels use the concept of M.I. to store rotational energy efficiently.

Importance of Moment of Inertia

  • It determines how easily or difficultly a body can rotate.
  • Higher M.I. means higher resistance to change in rotation.
  • It is a key factor in mechanical stability, dynamic balance, and vibration control.

Thus, understanding and calculating M.I. is crucial in designing safe and efficient mechanical systems.

Example

Example: Find the moment of inertia of a solid sphere of mass and radius .

Solution:

Hence, the moment of inertia of the sphere is 0.072 kg·m².

Conclusion

In conclusion, the moment of inertia (M.I.) is a fundamental property of rotating bodies that defines their resistance to angular acceleration. It depends on both the mass and its distribution about the axis of rotation. The M.I. plays an essential role in analyzing rotational dynamics, mechanical stability, and the design of machines and structures. A clear understanding of M.I. helps engineers optimize performance, reduce vibration, and enhance the safety and efficiency of mechanical systems.