A Centroid

Physics

Centroid of a Volume:

If the body is made from a homogeneous material, then its density \(\rho\) will be constant. Therefore, a differential element of volume dV has a mass \(dm =\rho dV\). Substituting this into

\(C_m(\frac{\int \hat x dm}{\int dm}, \frac {\int \hat y dm}{\int dm},\frac {\int \hat z dm}{\int dm})\) ( coordinates of the center of mass)

and canceling out r, we obtain formulas that locate the centroid C or geometric center of the body

\[C(\frac{\int \hat x dV}{\int dV}, \frac {\int \hat y dV}{\int dV},\frac {\int \hat z dV}{\int dV})\]

Centroid of an Area:

If an area lies in the x–y plane and is bounded by the curve y = f (x), then its centroid will be in this plane and can be determined from integrals

\[C(\frac{\int \hat x dA}{\int dA}, \frac {\int \hat y dA}{\int dA},0)\]

Dividing an Area into Differential Rectangular Strips helps in determining the centroid of thr plane area, the area can be divided into a large number of very thin rectangular strips. As the width of each strip approaches zero, each strip becomes a differential area element dA.

We can choose either vertical strips or horizontal strips, depending on which choice makes the integration simpler.

For a vertical strip:

dA = y dx

The centroid of the strip is located at:

x̃ = x

ỹ = y/2

For a horizontal strip:

dA = x dy

The centroid of the strip is located at:

x̃ = x/2

ỹ = y

The coordinates of the entire area's centroid are then obtained by taking the area-weighted average of the coordinates of all the differential strips:

x̄ = ∫ x̃ dA / ∫ dA

ȳ = ∫ ỹ dA / ∫ dA

Thus, by dividing the area into differential rectangular strips and summing their contributions through integration, we can determine the coordinates of the centroid of the entire area.

Centroid of a Line:

If a line segment (or rod) lies within the x–y plane and it can be described by a curve y = f (x), then its centroid is determined from

\[C(\frac{\int_L \hat x dL}{\int_L dL}, \frac {\int_L \hat y dL}{\int _LdL},0)\]

\[dL=\sqrt {(dx)^2+(dy)^2}\]

Physics

the geometric center of a shape or area.