Question

What is $1st$ and $2nd$ moment of area?

Answer 1

Hint: Arithmetic mean of area is usually utilized in engineering applications to see the centroid of an object or the statically moment of area. The moment of area, also called moment of inertia of plane area, or second area moment, may be a geometrical property of a region which reflects how.

Complete step by step answer:
First moment of area:The first moment of the area relies on the mathematical construct moments in metric spaces. It's a measure of the spatial distribution of a shape in reference to an axis. the primary moment of area of a shape, a few certain axis, equals the sum over all the infinitesimal parts of the form of the realm of that part times its distance from the axis \[\left[ {\Sigma \left( {a{\text{ }} \times {\text{ }}d} \right)} \right].\]

Given a vicinity, A, of any shape, and division of that area into $n$ number of very small, elemental areas , $d{A_i}$. Let ${x_i}$ and ${y_i}$ be the distances (coordinates) to every elemental area measured from a given x-y axis. Now, the primary moment of area within the x and y directions are respectively given by:
${S_x} = A\bar y = \sum\limits_{i = 1}^n {{y_i}{\mkern 1mu} d{A_i}} = \int\limits_A y dA$ and ${S_y} = A\bar x = \sum\limits_{i = 1}^n {{x_i}{\mkern 1mu} d{A_i}} = \int\limits_A x dA$ .

Second moment of area: The moment of area, or second area moment, or quadratic moment of area and also called the realm moment of inertia, may be a geometrical property of a locality which reflects how its points are distributed with respect to an arbitrary axis.
The moment of area is usually denoted with either an I (for an axis that lies within the plane) or with a J (for an axis perpendicular to the plane).

In both cases, it's calculated with a multiple integral over the item in question.
Uses: In structural engineering, the moment of area of a beam is a vital property employed in the calculation of the beam's deflection and therefore the calculation of stress caused by an instant applied to the beam.

Function:The moment of area, also called "moment of inertia of plane area", "polar moment of inertia", "area moment of inertia", or "second area moment", may be a property of a cross-section that may be accustomed predict the resistance of a beam to bending and deflection around an axis that lies within the cross-sectional plane.

Note: The primary moment of area of a shape, a couple of certain axis, equals the sum over all the infinitesimal parts of the form of the realm of that part times its distance from the axis $\left[ {\sum\limits_{}^{} {\left( {a \times d} \right)} } \right]$. The moment of area is denoted by I or J. In both cases, it's calculated with a multiple integral over the thing in question.