Centroids and Moments of Inertia Flashcards
What is a centroid?
The geometric center of a plane figure is the arithmetic mean (“average”) position of all the points in the shape.
How many first moments are there and what are they?
There are 2 first moments. The moment with respect to the x and the moment with respect to the y.
What is the first moment with respect to the y axis equal to?
The sum of the moments about the y axis. So the sum of the product of the subarea and the x distance to the point of rotation.
What is the centroid of a composite line segment in the x y plane?
The center a line composed of different line segments.
What is the center of a composite line equal to in the x direction?
The sum of the product of the average length of the line in the x direction and the total length of the line. Divided by the entire length of the composite line.
What is the centroid of an area in the x y plane?
The centroid of an area composed of subareas.
What is the center of area in the x direction equal to?
The moment with respect to the y axis divided by the entire area.
What is the center of area in the y direction equal to?
The moment with respect to the x axis divided by the entire area.
What is the centroid of volumes?
The centroid of a volume composed of sub volumes.
What is the centroid of volume equal to in the x y or z axis?
The sum of the product of the distance to the point of rotation and the volume of the sub volume. Divided by the entire volume.
What does a solid body contain in three dimensions?
It will have a centroid and a center of gravity.
Are the center of gravity and centroid the same for the solid body?
The locations of these two point do not always coincide.
What is the center of gravity?
The center of the mass of the object.
When are the center of gravity and centroid the same?
When the body is homogeneous.
Are the moments of inertia always positive?
Yes
Can the product of the moment of inertias be negative and why?
Yes because the x and y distances from the composite centroid to the subarea can be negative depending on where the centroid is located.
What is the moment of inertia?
A measure of a beams ability to resist bending.
How does a beam react to bending with a small moment of inertia?
It will bend more than a beam with a large moment of inertia.
Is the moment of inertia always positive?
Yes
What is the moment of inertia dependent on?
It is dependent on the orientation of the beam.
How many moment of inertias are there?
Two one with respect to the x axis and one with respect to the y axis.
What is the centroidal moment of inertia?
The moment of inertia taken with respect to an axis passing through the areas centroid.
What is the smallest possible moment of inertia for the area?
The centroidal moment of inertia.
What is the polar moment of inertia?
A measure of an areas resistance to torsion.
What is the perpendicular axis theorem?
The moment of inertia of a plane area about an axis normal to the plane is equal to the sum of the inertias about any mutual perpendicular axis lying in the plane and passing through the axis.
What moments of inertia are most convenient to use for the parallel axis theorem?
The centroidal moment of inertia.
What is the parallel axis theorem?
If the moment of inertia with respect to an axis is known the moment of inertia with respect to another parallel axis can be found.
When is this theorem used?
To evaluate the moment of inertia of areas that are composed of two or more basic shapes.
What is the radius of gyration?
Is an imaginary distance from the centroidal axis to at which the entire area can be assumed to exist without changing the moment of inertia.
What is the product of inertia?
The product of inertia for a two dimensional area is found by multiplying each differential element of area by its x and y coordinate and then summing over the entire area.
When is the product of inertia zero?
When either axis is an axis of symmetry.
Can the product of inertia be negative?
Yes.
What is the transfer theorem for the products of inertia equal to?
It is equal to the sum of the ce3ntroidal product of inertia of the old system plus the product of the x and y distances in the new coordinates to the new centroid and the area.
