What is a Moment of Inertia?
• It is a measure of an object’s resistance to
changes to its rotation.
• Also defined as the capacity of a cross-section to
resist bending.
• It must be specified with respect to a chosen axis
of rotation.
• It is usually quantified in mm4 , cm4 , m4 or
inch4 , kgm2
The moment of inertia about any point or axis is
the product of the area and the perpendicular
distance between the point or axis to the centre of
gravity of the area. This is called the first moment
of area.
If this first moment of area is again multiplied by
the perpendicular distance between them, the
product is known as second moment of area.
In case of a rigid body, its mass is considered and it
is called second moment of mass.
Second moment of area = Moment of Inertia
Second moment of mass = Mass Moment of Inertia
Perpendicular Axis Theorem
•The moment of inertia (MI) of a plane area
about an axis normal to the plane is equal to
the sum of the moments of inertia about any
two mutually perpendicular axes lying in the
plane and passing through the given axis.
•That means the Moment of Inertia Iz = Ix+Iy
Parallel Axis Theorem
•The moment of area of an object about any
axis parallel to the centroidal axis is the sum of
MI about its centroidal axis and the prodcut of
area with the square of distance from the
reference axis.
•Essentially, IXX= IG+Ad2
•A is the cross-sectional area. d is the
perpendicuar distance between the centroidal
axis and the parallel axis
Parallel axes theorem
Problem: Determine the moment of inertia of the T-section shown in
Figure below with respect to its centroidal Xo axis.
Solution:
A hollow square cross section consists of an 8 in. by 8 in. square from
which is subtracted a concentrically placed square 4 in. by 4 in. Find
the polar moment of inertia and the polar radius of gyration with
respect to a z axis passing through one of the outside corners.
Problem: Find the moment of inertia Ix of the body about centroidal
axis.
Procedure: First find the MI of the whole rectangle
(120mm*180mm) and then subtract the MI of the
white rectangle (120mm*80mm) from the total area.
Problem: Determine the moment of inertia of the shaded area shown with
respect to each of the coordinate axes.
Problem:
Problem: