MOMENT OF INERTIA

• MOMENT OF INERTIA: - The moment of a force about any points is known as first moment of the force. When the first moment of the force is multiplied again by the distance, the product is called second moment of the force or moment of moment of the force.

        If area is considered in the place of force, then it is called area MI.

        If mass is considered in the place of force, then it is called mass MI.

        (NOTE: - MI= Moment of Inertia)

• UNIT: - For Mass MI, SI unit is kg-m²

                    For area MI, SI unit is m⁴

        Mass MI of a body is considered as the measure of resistance of the body to rotation.

        Area MI of a body is considered as the measure of resistance of the body against bending.

 

• MOMENT OF INERTIA BY ROUTH’S LAW: -

    If a body is symmetrical about three mutually perpendicular axes, then the MI about any one axis passing through cg is

    I= [A (or M) X S]/3  for a square or rectangular lamina.

    I= [A (or M) X S]/4  for a circular or elliptical lamina.

    I= [A (or M) X S]/5  for a spherical body.

[NOTE: - “S” is the sum of the squares of the two semi-axis, other than the axis about which MI is required to found out.

A= Area of the lamina.

M= Mass of the body.]

 

• PARALLEL AXIS THEOREM: -

        If the moment of inertia of a plane or area about an axis through its CG is denoted by I_g, then the MI of the area about any other axis parallel to the first axis is equal to the sum of MI about CG axis and the product of area (or mass) and square of the distance between the axis.

          For lamina I_ab = I_g+ah²

Fig.1 | Parallel Axis Theorem

        For body I_ab = I_g+mh²

• PERPENDICULAR AXIS THEOREM: -

        MI of an area about any axis perpendicular to the plane of the area at any point is equal to the sum of MI about any two mutually perpendicular axes through the same point in the plane of the area.

Fi. 2 | Parallel Axis Theorem

        

• MI OF A RECTANGULAR SECTION: -

Fig. 3 | MI of a Rectangular section

• MI OF A TRIANGULAR SECTION: -

Fig. 4 | MI of a Triangular section

• MI OF A CIRCULAR, SEMI CIRCULAR, QUADRANT OF A CIRCULAR SECTION: -

Fig.5 | MI of Circular, Semi circular, quadrant of a circle

• MI OF A  SPHERE: -

Fig.6 | MI of a Sphere

• MI OF A  CIRCULAR RING OR HOLLOW CYLINDER: -

Fig.7 | MI of a circular ring

•  MI OF A SQUARE :-

Fig.8 | MI of a Square section

• MI OF A SOLID CONE: -

Fig.9 | MI of Solid Cone