PARTICLE + RIGID BODIES
The term particles does not imply small corpuscles; it implies that the size and shape of the bodies under consideration will not significantly affect the solution of the problems. The term rigid bodies implies bodies that does not deform.

EQUILIBRIUM OF RIGID BODIES
∑ Fx = 0
∑ Fy = 0
∑ Fz = 0
∑ Mx = 0
∑ My = 0
∑ Mz = 0

FREE BODY DIAGRAM OF RIGID BODIES
below is a simple example :


CENTER OF GRAVITY
center of gravity = center of weight
the following 2 equations are meant for 2D object :
X     = ( ∫xdW ) Wtot     = ( ∑xiWi ) Wtot
Y     = ( ∫ydW ) Wtot     = ( ∑yiWi ) Wtot
the following 3 equations are meant for 3D object :
X     = ( ∫xdW ) Wtot     = ( ∑xiWi ) Wtot
Y     = ( ∫ydW ) Wtot     = ( ∑yiWi ) Wtot
Z     = ( ∫zdW ) Wtot     = ( ∑ziWi ) Wtot

CENTROID
centroid = center of geometry
the following 2 equations are meant for 2D object :
X     = ( ∫xdA ) Atot     = ( ∑xiAi ) Atot
Y     = ( ∫ydA ) Atot     = ( ∑yiAi ) Atot
the following 3 equations are meant for 3D object :
X     = ( ∫xdV ) Vtot     = ( ∑xiVi ) Vtot
Y     = ( ∫ydV ) Vtot     = ( ∑yiVi ) Vtot
Z     = ( ∫zdV ) Vtot     = ( ∑ziVi ) Vtot

HOMOGENEOUS MATERIAL
first of all, please recall the fundamental animals as below :
weight, W = m × g
density, ρ = M V
if the material of an object is homogeneous, it is said the ρ is constant
thus, center of gravity = centroid

FIRST MOMENT OF AN AREA
first moment of A wrt x-axis, Qx = ∫ydA = ∑yiAi
first moment of A wrt y-axis, Qy = ∫xdA = ∑xiAi

MOMENT OF INERTIA OF AN AREA
second moment is also known as moment of inertia
second moment of A wrt x-axis, Ix = ∫y2dA
second moment of A wrt y-axis, Iy = ∫x2dA
polar moment of inertia of A wrt origin (o), Jo = ∫r2dA
MOMENT OF INERTIA OF A MASS
I = ∫r2dm
PARALLEL-AXIS THEOREM OF AN AREA
Inew = Icentroidal + Ad2
PARALLEL-AXIS THEOREM OF A MASS
Inew = Icentroidal + md2
RADIUS OF GYRATION OF AN AREA
k = ( I / A )
RADIUS OF GYRATION OF A MASS
k = ( I / m )


COMMON SHAPES : CENTROID
below are some selections :


COMMON SHAPES : MOMENT OF INERTIA OF AN AREA
below are some selections :


COMMON SHAPES : MOMENT OF INERTIA OF A MASS
below are some selections :


SHEAR & BENDING MOMENT IN A BEAM
beams are treated as rigid bodies in Vector Mechanics (Statics)
first of all, let's review the sign conventions :

below is a simple example :


MOHR'S CIRCLE
C = [ (Ix+Iy)/2 , 0 ]
X = [ Ix , Ixy ]
Y = [ Iy , -Ixy ]
X' = [ Ix' , Ix'y' ]
Y' = [ Iy' , -Ix'y' ]


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