Why Is y Bar Zero in the Parallel Axis Theorem for Area?

[goldfish9776]

Homework Statement

why the y bar is 0 ? according to the diagram , y ' has certain value , it's not 0 ! can someone help to explain ?

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

$\overline{y}= 0$ because the y' axis, not the y axis, is defined as the vertical line passing through the center of mass.

 

[goldfish9776]

$\overline{y}= 0$ because the y' axis, not the y axis, is defined as the vertical line passing through the center of mass.

it pass thru centroid , why it is 0 ?

 

[haruspex]

it pass thru centroid , why it is 0 ?

The shape can be thought of as made up of many little areas like A. Each little area has its own (x', y') coordinates relative to the centroid. y-bar is here defined as the average value of y' across all these little areas. By definition of centroid, that average is zero.

 

[goldfish9776]

The shape can be thought of as made up of many little areas like A. Each little area has its own (x', y') coordinates relative to the centroid. y-bar is here defined as the average value of y' across all these little areas. By definition of centroid, that average is zero.

why only the second integral = 0 , why not the first integral equal to 0 also ?

 

[SteamKing]

why only the second integral = 0 , why not the first integral equal to 0 also ?

Because in the first integral, y' is squared.

When y' is by itself, you are adding products of y' dA on either side of the centroidal axis, so some products are negative and some are positive. When you calculate y'² dA, all of the products are positive, so their sum adds up to a positive, non-zero result.