Centroid and Center Of Mass of the Composite Bodies
What is center of gravity?
The center of gravity G is a point which locates the resultant weight of a system of particles.
What is composite body?
A composite body consists of a series of connected simplershaped bodies, which may be rectangular, triangular, semicircular, etc.
What is centroid?
The line of action of the resultant force passes through the geometric center or centroid of the volume under the distributed–loading diagram.
Find the Centroid of A Rectangular Area by integration
Find the centroid of a quarter circle by double integration in rectangular coordinates
Procedure for locatingcenter of gravity of a body or the centroid of a composite geometrical object
First Moments of Areas and Lines
 
 

Center Of Mass – locates the point in a system where the resultant mass can be concentrated so that the moment of the concentrated mass with respect to any axis equals the moment of the distributed mass with respect to the same axis.
Center Of Gravity– locates where the resultant, concentrated weight acts on a body.
Centre of gravity and centroid:
The center of gravity (G) is a point that locates the resultant weight of a system of particles.
Particles with weights W_{1}, W_{2}, …, W_{n} can be replaced by a resultant force of weight W located at the center of gravity G.
To find the location of the center of gravity G(x,y,z):
(We can obtain z by imagining the coordinate system, with the particles fixed in it, as being rotated 90 degrees about the x (or the y) axis).
Knowing W=mg, if we assume that the acceleration due to gravity (g) for every particle is constant (g will be cancelled out)
A rigid body is composed of an infinite number of particles, hence it is necessary to use integration instead of summation.
But dm=ρdV, with ρ being the density and dv the volume of each particle. Therefore, the centre of mass has the coordinates of
Centroid:
The centroid (C) is a point which defines the geometric center of an object. If the material composing a body is uniform or homogeneous, the density of material is constant (ρ = constant). Hence, the resulting formulas that define the centroid of a body depend only on the geometry of the body {Volume (V), Area (A), or Length (L)}.
Centroid (volume):
Centroid (area):
Centroid (line):
The centroids of some shapes may be partially or completely specified by using conditions of symmetry. In cases where the shape has an axis of symmetry, the centroid of the shape will lie along that axis.