Which of the following option is true about the moment of inertia of a section?
i) The unit of moment of an area does not depends upon the units of its area.
ii) The moment of inertia of an area may be obtained by the methods of integration.
iii) The Routh's rule is used in finding out the moment of a body which is unsymmetrical about three mutually perpendicular axes.
The true option is

Explanation:

• The moment of inertia of an area represents how that area is distributed relative to a reference axis. It reflects the section's ability to resist bending — the larger the value, the stiffer the section.

• It can be calculated using integration methods by summing up the contributions of small elemental areas across the entire section. This is particularly useful for irregular or complex shapes where standard formulas are not applicable.

• The unit of moment of inertia depends on the units of both area and distance. If area is measured in square meters (m²), and distance in meters (m), then the unit of moment of inertia becomes meters to the fourth power (m⁴). Thus, units of area and distance both influence the unit of moment of inertia.

• Moment of inertia of an area (second moment of area) is different from mass moment of inertia, which concerns the distribution of mass and is used in rotational dynamics. Concepts like Routh’s Rule apply to mass moment of inertia, not to second moment of area.

 Additional Information Routh's Rule

• Routh’s Rule is a method used in dynamics to determine the mass moment of inertia of a rigid body about an arbitrary axis. It is particularly helpful when the body is unsymmetrical and does not align neatly with standard coordinate axes.

• It is based on the known mass moments of inertia about three mutually perpendicular axes that pass through a common point (usually the center of mass or centroid of the body).

• The rule allows engineers to calculate the moment of inertia about any axis that does not coincide with these principal axes, by combining the known moments of inertia and considering the orientation of the new axis.

• Routh’s Rule is used mainly in mechanical and aerospace engineering, where bodies rotate about complex axes (such as in gyroscopes, propellers, and unsymmetrical machinery), and helps in dynamic analysis and design of rotating systems.