moment of inertia
Moment of Inertia
Academic Resource Center
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 m4 or kgm2
Quick Note about Centroids….
• The centroid, or center of gravity, of any object is the point within that object from which the force of gravity appears to act. • An object will remain at rest if it is balanced on any point along a vertical line passing through its center of gravity.
and more…
• The centroid of a 2D surface is a point that corresponds to the center of gravity of a very thin homogeneous plate of the same area and shape.
• If the area (or section or body) has one line of symmetry, the centroid will lie somewhere along the line of symmetry.
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 of 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 Axis Theorem - Derivation
• Consider the moment of inertia Ix of an area A with respect to an axis AA’. Denote by y, the distance from an element of area dA to AA’.
I x y dA
2
Derivation (cont’d)
• Consider an axis BB’ parallel to AA’ through the centroid C of the area, known as the centroidal axis. The equation of the moment inertia becomes: I x y dA y
Academic Resource Center
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 m4 or kgm2
Quick Note about Centroids….
• The centroid, or center of gravity, of any object is the point within that object from which the force of gravity appears to act. • An object will remain at rest if it is balanced on any point along a vertical line passing through its center of gravity.
and more…
• The centroid of a 2D surface is a point that corresponds to the center of gravity of a very thin homogeneous plate of the same area and shape.
• If the area (or section or body) has one line of symmetry, the centroid will lie somewhere along the line of symmetry.
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 of 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 Axis Theorem - Derivation
• Consider the moment of inertia Ix of an area A with respect to an axis AA’. Denote by y, the distance from an element of area dA to AA’.
I x y dA
2
Derivation (cont’d)
• Consider an axis BB’ parallel to AA’ through the centroid C of the area, known as the centroidal axis. The equation of the moment inertia becomes: I x y dA y