Centroids and Moments of Inertia

Question 1

What is a centroid?

Answer 1

The geometric center of a plane figure is the arithmetic mean (“average”) position of all the points in the shape.

Question 2

How many first moments are there and what are they?

Answer 2

There are 2 first moments. The moment with respect to the x and the moment with respect to the y.

Question 3

What is the first moment with respect to the y axis equal to?

Answer 3

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.

Question 4

What is the centroid of a composite line segment in the x y plane?

Answer 4

The center a line composed of different line segments.

Question 5

What is the center of a composite line equal to in the x direction?

Answer 5

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.

Question 6

What is the centroid of an area in the x y plane?

Answer 6

The centroid of an area composed of subareas.

Question 7

What is the center of area in the x direction equal to?

Answer 7

The moment with respect to the y axis divided by the entire area.

Question 8

What is the center of area in the y direction equal to?

Answer 8

The moment with respect to the x axis divided by the entire area.

Question 9

What is the centroid of volumes?

Answer 9

The centroid of a volume composed of sub volumes.

Question 10

What is the centroid of volume equal to in the x y or z axis?

Answer 10

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.

Question 11

What does a solid body contain in three dimensions?

Answer 11

It will have a centroid and a center of gravity.

Question 12

Are the center of gravity and centroid the same for the solid body?

Answer 12

The locations of these two point do not always coincide.

Question 13

What is the center of gravity?

Answer 13

The center of the mass of the object.

Question 14

When are the center of gravity and centroid the same?

Answer 14

When the body is homogeneous.

Question 15

Are the moments of inertia always positive?

Answer 15

Yes

Question 16

Can the product of the moment of inertias be negative and why?

Answer 16

Yes because the x and y distances from the composite centroid to the subarea can be negative depending on where the centroid is located.

Question 17

What is the moment of inertia?

Answer 17

A measure of a beams ability to resist bending.

Question 18

How does a beam react to bending with a small moment of inertia?

Answer 18

It will bend more than a beam with a large moment of inertia.

Question 19

Is the moment of inertia always positive?

Answer 19

Yes

Question 20

What is the moment of inertia dependent on?

Answer 20

It is dependent on the orientation of the beam.

Question 21

How many moment of inertias are there?

Answer 21

Two one with respect to the x axis and one with respect to the y axis.

Question 22

What is the centroidal moment of inertia?

Answer 22

The moment of inertia taken with respect to an axis passing through the areas centroid.

Question 23

What is the smallest possible moment of inertia for the area?

Answer 23

The centroidal moment of inertia.

Question 24

What is the polar moment of inertia?

Answer 24

A measure of an areas resistance to torsion.

Question 25

What is the perpendicular axis theorem?

Answer 25

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.

Question 26

What moments of inertia are most convenient to use for the parallel axis theorem?

Answer 26

The centroidal moment of inertia.

Question 27

What is the parallel axis theorem?

Answer 27

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.

Question 28

When is this theorem used?

Answer 28

To evaluate the moment of inertia of areas that are composed of two or more basic shapes.

Question 29

What is the radius of gyration?

Answer 29

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.

Question 30

What is the product of inertia?

Answer 30

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.

Question 31

When is the product of inertia zero?

Answer 31

When either axis is an axis of symmetry.

Question 32

Can the product of inertia be negative?

Answer 32

Yes.

Question 33

What is the transfer theorem for the products of inertia equal to?

Answer 33

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.