To find \, we have to divide the whole Figure into standard areas.Įxample: Find the moment of inertia of a plate with a circular hole about its centroidal x axis as shown in Fig.8. Sol: The given I-section is symmetrical about the y-y axis, therefore, = + Įxample: Determine the moment of inertia about the horizontal axis passing through the centroid of the section as shown in Fig.7. (d) Moment of Inertia of the triangular section about an axis passing through its centroid and parallel to baseįig.\\] (c) Moment of Inertia of a rectangular sectionįig. 3. Moment of inertia of circular section.įig. 2.Moment of inertia of a triangular section about an axis passing through its centroid and parallel to base. Give equation for the following by explaining each term used in that equation :- 1.Moment of inertia of a rectangular section. Let (r) be the distance of the lamina (P) from Z-Z axis such that. Now consider an axis OZ perpendicular to OX and OY. The second moment of area about x-axis and y-axis can be found by integrating the second moment of area of small. Consider a small lamina of area A as shown in Figure 11.14. Second moment of area is also known as area moment of inertia.
Proof : Consider a small lamina (P) of area having co-ordinates as X and Y along two mutually perpendicular OX and OY on a plane section as shown in fig. Thus, the centroid of a triangle is at a distance h/3 from the base and 2h/3 from the apex where h is the height of the triangle. Hint: Hint: first of all, find the second moments of area of a triangle about any of its base. Polar moment of inertia is also denoted as. center of the circular whole is exactly at the centroid of the inclined rectangular part. Z-Z axis is called polar axis and is known as the polar moment of inertia. Where X-X and Y-Y are two mutually perpendicular axis in the plane of the lamina and Z-Z is an axis passing through the centroid and perpendicular to the plane of the lamina. The perpendicular axis theorem states that the moment of inertia of a plane figure about an axis perpendicular to the figure and passing through the centroid is equal to the sum of moment of inertia of the given figure about two mutually perpendicular axis passing through the centroid and lying in the given figure. State and prove theorem of perpendicular axis. The moment of inertia of the whole lamina about AB is given by Suppose we have an object, and we want to find its moment of inertia around some. Moment of inertia of the area about AB is given by This is, of course, a vector equation which is really three equations. Consider an elemental area at a distance from the line. It states that the moment of inertia of a lamina about any axis in the plane of the lamina is equal to the sum of the Moment of Inertia of that lamina about its centroidal axis parallel to the given axis and the product of the area of the lamina and square of the perpendicular distance between the two axis. State and prove theorem of parallel axis. The perpendicular axis theorem states that the moment of inertia of a plane figure about an axis perpendicular to the figure and passing through the centroid is equal to the sum of moment of inertia of the given figure about two mutually perpendicular axis passing through the centroid and lying in the plane of the given figure. It states that the moment of Inertia of a lamina about any axis in the plane of the lamina is equal to the sum of the Moment of Inertia of that lamina about its centroidal axis parallel to the given axis and the product of the area of the lamina and the square of the perpendicular distance between the two axis. The distance of a point where the whole area of a body is assumed to be concentrated from a given axis is called radius of gyration.
It is called second moment of area because we are taking moment of area about an axis twice. of the area from that axis is known as second moment of area. The product of the area and the square of the distance of the C.G. Moment of inertia is also termed as the second moment of mass and is denoted by I.