Centre of Gravity and Moment of Inertia

Centre of gravity: It is a defined as the point through which resultant of force of gravity (weight) of the body acts.

• C. G. of a Flat plate:
  

Centroid: If the thickness of a solid body (say plate) reduces to infinitesimal, then the plate reduces to an area. Centroid is a point in a plane area such that the moment of area about any axis

through that point is zero.

W. r. to any reference point,

Difference between centre of gravity and Centroid:

• The term centre of gravity applies to bodies with mass and weight, and centroid applies to plane area.
• Centre of gravity of a body is a. point through which the resultant gravitational force (weight) acts for any orientation ofthe body where as centroid is a point in a plane are such that the moment of area about any axis through that point is zero.

Use of axis of symmetry: Centroid of an area lies on the axis of symmetry, if it exists.

Eg:

• Centroid of a circle is its centre.
• Centroid of a rectangle of sides ‘b’ and ‘d’ is at a distance ‘b/2’ and ‘d/2’ from one corner.
  

Centroid of some common figures :

Centroid of composite sections (Say I, T and L sections):

Step1: Split the given composite section into suitable standard figures and find the centroid of each of them.

Step 2: Assuming the area of simple figure as concentrated at its centroid, its moment about

an axis can be found by multiplying the area with distance of its centroid from the reference axis.

Step 3: After determining the moment of each area about reference axis, the distance of centroid from the axis is obtained by dividing total moment of area by total area of the composite section.

Moment of Inertia:

Principal axes : Are defined as those about which the product of Inertia is zero.

Principal Moments of Inertia: Are the M . I for which the product of inertia is zero.

The principal axis are those for which,

The maximum and minimum moments inertia will be about the principal axes.

Principal axes can be calculated by

Where α = angle of major principal axis with horizontal

α1= α + 90 = angle of minor principal axis with horizontal

Note: Ix + Iy = Ix' + Iy'

i.e.,the sum of moments of Inertia around  two mutually Perpendicular  centroidal  axes is invariant(constant).

Parallel Axes Theorem: The moment of inertia of a plane area with respect to any axis in its plane is equal to the moment of inertia with respect to a parallel centroid axis plus the product of the total area and the square of the distance between the two axes.

If the moment of inertia of a plane area about an axis through its e.g. is IGG , the moment of inertia of the area about an axis AB parallel to the first and at a distance ‘r’ from it, is given by

 IAB = IGG +Ar2

Perpendicular Axes Theorem: (polar moment of inertia) : If Ixx and Iyy be the moments of inertia of a plane section about two mutually perpendicular axes meeting at O, the moment of inertia Izz, i.e., about an axis perpendicular to both XX and YY passing

through their point of intersection is given by

Izz = lxx + Iyy

• Izz is called polar moment of inertia
• Solid circular section IZ = J = πd4/ 32
• Hollow circular section
• Square is IZ = a4/6

Radius of gyration: Radius of gyration is the common distance at which the elemental areas may be placed, so that the moment of inertia of the whole section is not changed.

r = (I/A)1/2

r = Radius of gyration is in mm or cm.

• r for a solid circle is d/4
• r for a square is, a/2(3) 1/2

Section Modulus (Z): Modulus of section or section modulus is the ratio of moment of inertia about its e.g. to the distance of the extreme fibre from the centroidal axis.

• Section modulus has units mm3 or cm3.

Moment of Inertia of Common fig: