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In Mathematics , the Centroid is the point at which the region/volume is equally balanced.


FINDING THE CENTER OF MASS


Prerequisites


Moments


First, picture an unbalanced see-saw. The see-saw will only balance if the mass of the first person times their distance form the center is equal to the mass of the other person times his/her distance from the center. This is a concept from physics stated in Archimedes' law of the lever. Assuming this we can find the center of complicated figures. Let's call the quantity m_1 x_1 the moment of the mass m_1 with respect to the origin. Clearly if one is to place multiple people on the see-saw the sum of their moments must be equal, so let's try to use this fact to find the center of mass of an object. Let's use the number line, if one object is at k (to the left) and the other is at r with masses m_1 and m_2 respectively it is easy to see that

: m_1 ( \mathrm{origin} - k ) = m_2 ( r - \mathrm{origin} ).

Solving for origin we have

: m_1 ( \mathrm{origin} - k ) = m_2 ( r - \mathrm{origin} ) = m_1 \mathrm{origin} + m_2 \mathrm{origin} = m_1 k + m_2 r

which takes us further to say that

: rac{m_1 x + m_2 r }{m_1 + m_2} = \mathrm{origin}.

We see that the above equation is very easy to generalize so we have

: rac{m_1 x + m_2 r }{m_1 + m_2} = \mathrm{origin} = rac{m_1 x_1 + m_2 x_2 + \cdots}{m_1 + m_2 + \cdots} = rac{ \sum_{k=1}^n m_k x_k }{ \sum_{k=1}^n m_k }.

Using this we can find what is called the moment about the origin.


Discrete moments in two dimensions

This is easily extendable to two dimensions, just think of the lamina or plane on which the masses sit as a two dimensional see-saw, so we only have to find the moment about the y-axis. To find both the x and y coordinates of the center we will use the perpendicular distance to the x-axis and y-axis respectively. Make sure that you use the signed distances to the ''x'' or ''y'' axis, i.e. the point (1, −2) should be considered to have a ''y''-coordinate or −2. Commonly the notation for the center of a region is denoted \left ( M_x, M_y, M_z , \ldots, ight )


The center of a planar lamina




Note the integrals can be done in any order as that is guaranteed by Fubini's Theorem.


The center of a 3-dimensional object with variable density

Instead of finding the perpendicular distance to a line we find it to a plane, specifically the ''xy''-, ''yz''-, and ''xz''-planes. The derivation is the same as for a Lamina with variable density with only an extra dimension.

: \mathrm{center}_x = rac{M_x}{\mathrm{total\ mass}} = rac{\int_{a_2}^{b_2} \int_{a_1}^{b_1} \int_{a_3}^{b_3} x \,\mathrm{density}(x,y,z) dz\, dy\, dx}{\int_{a_2}^{b_2} \int_{a_1}^{b_1} \int_{a_3}^{b_3} \mathrm{density}(x,y,z)\, dz\, dy\, dx}

and

: \mathrm{center}_y = rac{M_y}{\mathrm{total\ mass}} = rac{\int_{a_2}^{b_2} \int_{a_1}^{b_1} \int_{a_3}^{b_3} y \,\mathrm{density}(x,y,z)\, dz\, dy\, dx}{\int_{a_2}^{b_2} \int_{a_1}^{b_1} \int_{a_3}^{b_3} \mathrm{density}(x,y,z)\, dz\, dy\, dx}

: \mathrm{center}_z = rac{M_z}{\mathrm{total\ mass}} = rac{\int_{a_2}^{b_2} \int_{a_1}^{b_1} \int_{a_3}^{b_3} z \,\mathrm{density}(x,y,z)\, dz\, dy\, dx}{\int_{a_2}^{b_2} \int_{a_1}^{b_1} \int_{a_3}^{b_3} \mathrm{density}(x,y,z)\, dz\, dy\, dx}