Find M.I. of a circle about its diameter.

Short Answer:

The moment of inertia (M.I.) of a circle about its diameter represents the measure of the area’s resistance to bending or rotation about that axis. For a circular section of radius , the M.I. about its diameter (either the x-axis or y-axis) is given by the formula:

If the circle’s diameter is denoted as , the same formula can also be written as:

These values are useful in designing circular beams, shafts, and other rotational components in mechanical and structural engineering.

Detailed Explanation :

Moment of Inertia of a Circle about its Diameter

In Engineering Mechanics, the moment of inertia (M.I.) is the property of a body that determines its resistance to bending or rotational motion about an axis. When we consider a circle, the M.I. about its diameter is an important quantity used in analyzing stresses and deflections in circular beams, discs, and other cylindrical structures.

A circular area is symmetrical about both its diameters; therefore, the moment of inertia about any diameter is the same. This property makes calculations simpler and helps in determining the bending behavior and torsional strength of circular components.

Definition

The moment of inertia of a circle about its diameter can be defined as:
“The sum of the products of each small elemental area and the square of its perpendicular distance from the diameter.”

Mathematically,

or

where,

  • = elemental area,
  • = perpendicular distances of the element from the respective axes.

Due to symmetry of the circle, .

Derivation of Moment of Inertia of a Circle about its Diameter

Consider a circle of radius centered at the origin, lying in the -plane. We have to find the M.I. about the x-axis (or diameter) passing through the center.

Equation of a circle is:

Take an elemental horizontal strip of thickness at a distance from the x-axis.
The length of the strip = .

Hence,

Now, the M.I. of this strip about the x-axis is:

Integrating this over the whole circle (from to ):

Because the circle is symmetrical,

Let , so .
Then, .

Substitute into the integral:

 

Using the trigonometric identity ,

Now,

Applying the limits to :

Therefore,

Since the circle is symmetrical,

Alternate Formula in Terms of Diameter

If the diameter of the circle is , then:

Hence,

This form is often used in mechanical engineering handbooks and structural analysis calculations.

Physical Meaning

The moment of inertia shows how the area is distributed relative to the diameter. A larger M.I. means the area is farther distributed from the axis, providing higher resistance to bending.
For circular shapes:

  • The symmetry ensures equal bending resistance in any direction.
  • It is particularly useful in rotating systems (shafts, discs, flywheels) where uniform strength in all directions is important.

Practical Importance

  1. Beam and Shaft Design:
    Used to calculate bending stresses () and deflections in circular cross-sections.
  2. Structural Engineering:
    Applied to analyze circular columns and arches that resist bending equally in all directions.
  3. Machine Elements:
    Useful in designing flywheels, pulleys, and rotors where uniform distribution of stress is required.
  4. Moment Distribution:
    Helps understand how load distribution varies when circular members are subjected to bending or twisting.
  5. Dynamic Analysis:
    Forms the base for calculating polar moment of inertia, which determines torsional strength and angular acceleration in rotating machinery.

Example

Example:
Find the moment of inertia of a circular plate of radius about its diameter.

Solution:

 

Hence,

This shows that the circular section has equal resistance about both diameters.

Key Points

  • due to symmetry.
  • Units: or (since it is an area moment of inertia).
  • The formula applies only for a solid circle.
  • For a hollow circle, the M.I. is .
Conclusion

In conclusion, the moment of inertia of a circle about its diameter is an essential parameter that determines its resistance to bending and rotational motion. For a solid circle, the M.I. about the diameter is given by or equivalently . The circular shape offers uniform bending strength in all directions, making it highly effective for use in shafts, pipes, and rotating components. Understanding this property is crucial for efficient and safe mechanical design.