equation for polar moment of inertia circle

If the area is composed of a collection of basic shapes whose centroidal locations are known with respect to the axis of interest, then the first moment of the composite area can be calculated as: The values x and y indicate the locations with respect to the axis of interest of the infinitesimally small areas, dA, of each element as the integration is performed. If you compare the equations for Q above to the equations for calculating the centroid (discussed in a previous section), you will see that we actually use the first moment of area when calculating the centroidal location with respect to an origin of interest.

$equation for polar moment of inertia circle$

Recall that the shear stress at any point located a distance y 1 from the centroid of the cross section is calculated as: The first moment is also used when calculating the value of shear stress at a particular point in the cross section. Where Q is the first moment of the area between the point y 1 and the extreme fiber (top or bottom) of the cross section. We are interested in calculating the shear stress at a point located at a distance y 1 from the centroid of the cross section. We may calculate the first moment of the area either above or below this location. In this case the point of interest is above the neutral axis, so it is simpler to consider the upper area which is shaded in blue in the figure below. This area extends from the point y 1 to the extreme fiber at the top of the cross section. It should be noted that the first moment of an area is either positive or negative depending on the position of the area with respect to the axis of interest. Therefore, the first moment of the entire area of a cross section with respect to its own centroid is zero.

The second moment of area, more commonly known as the moment of inertia, I, of a cross section is an indication of a structural member's ability to resist bending. Refer to (Figure) for the moments of inertia for the individual objects. In both cases, the moment of inertia of the rod is about an axis at one end. In (b), the center of mass of the sphere is located a distance R from the axis of rotation. In (a), the center of mass of the sphere is located at a distance L+R from the axis of rotation. Since we have a compound object in both cases, we can use the parallel-axis theorem to find the moment of inertia about each axis. The radius of the sphere is 20.0 cm and has mass 1.0 kg. The rod has length 0.5 m and mass 2.0 kg. Find the moment of inertia of the rod and solid sphere combination about the two axes as shown below.

Equation for polar moment of inertia circle