Moment of Inertia:
The moment of a force (P) about a point is the product of the force and perpendicular distance (x) between the point and the line of action of the force (P.x). This moment is also called the first moment of force. SupposeSupposeSupposeSuppose this moment is again multiplied by the perpendicular distance (x) between the point and the line of action of the force i.e, P.x(x) = Px2. In that case, that case, that case, that case, this quantity is called the moment of the moment of a force or second moment of force or moment of inertia (M.I).
Moment of Inertia of a Plane Area:
Consider a plane area whose moment of inertia must be found. Split up the whole area into several small elements.
Let, a1, a2, a3,… = Areas of small elements
r1, r2, r3,…= Corresponding distances of the elements from the line about which the moment of inertia is required to be found out.
Now, the moment of inertia of the area,
I = a1r12 + a2r22 + a3r32 +…
= ∑ar2
Unit of Moment of Inertia:
the unit of moment of inertia of a plane area depends upon the area’s units and the length.
1. If the area is in m2 and the length is also in m, the moment of inertia is expressed in m4.
2. If the area is in mm2 and the length is also in mm, the moment of inertia is expressed in mm4.
Method for Moment of Inertia:
The moment of inertia of a plane area may be found by any one of the following two methods:
- By Routh’s Rule
- By Integration
Moment of Inertia by Routh’s Rule:
Routh’s rule states that if a body is symmetrical about three mutually perpendicular axes, then the moment of inertia about anyone axis passing through its centre of gravity is given by:
I = A(M) x S / 3 … For a Square or Rectangular Lamina
I = A(M) x S / 4 … For a Circular or Elliptical Lamina
I = A(M) x S / 5 … For a Spherical Body
where,
A = Area of the plane area
M = Mass of the body
S = Sum of the squares of the two semi-axis, other than the axis about which the moment of inertia is required to be found out.
Moment of Inertia by Integration:
The moment of inertia of an area may also be found out by the method of integration as given below:
Consider a plane figure, whose moment of inertia is required to be found out about the X-X axis and Y-Y axis as shown below figure. Let us divide the whole area into several strips. Consider one of these strips.
Let, dA = Area of the strip
x = Distance of the centre of gravity of the strip on the X-X axis
y = Distance of the centre of gravity of the strip on the Y-Y axis
We know that the moment of inertia of the strip about the Y-Y axis
= dA.x2
Now the moment of inertia of the whole area may be found by integrating the above equation, i.e,
IYY = ∑dA.x2
Similarly, IXX = ∑dA.y2