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
∑ F_x = 0
∑ F_y = 0
∑ F_z = 0
∑ M_x = 0
∑ M_y = 0
∑ M_z = 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 ) ÷ W_tot     = ( ∑x_iW_i ) ÷ W_tot
Y     = ( ∫ydW ) ÷ W_tot     = ( ∑y_iW_i ) ÷ W_tot
the following 3 equations are meant for 3D object :
X     = ( ∫xdW ) ÷ W_tot     = ( ∑x_iW_i ) ÷ W_tot
Y     = ( ∫ydW ) ÷ W_tot     = ( ∑y_iW_i ) ÷ W_tot
Z     = ( ∫zdW ) ÷ W_tot     = ( ∑z_iW_i ) ÷ W_tot

CENTROID
centroid = center of geometry
the following 2 equations are meant for 2D object :
X     = ( ∫xdA ) ÷ A_tot     = ( ∑x_iA_i ) ÷ A_tot
Y     = ( ∫ydA ) ÷ A_tot     = ( ∑y_iA_i ) ÷ A_tot
the following 3 equations are meant for 3D object :
X     = ( ∫xdV ) ÷ V_tot     = ( ∑x_iV_i ) ÷ V_tot
Y     = ( ∫ydV ) ÷ V_tot     = ( ∑y_iV_i ) ÷ V_tot
Z     = ( ∫zdV ) ÷ V_tot     = ( ∑z_iV_i ) ÷ V_tot

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, Q_x = ∫ydA = ∑y_iA_i
first moment of A wrt y-axis, Q_y = ∫xdA = ∑x_iA_i

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 )½	RADIUS OF GYRATION OF A MASS k = ( I / m )½

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 = [ (I_x+I_y)/2 , 0 ]
X = [ I_x , I_xy ]
Y = [ I_y , -I_xy ]
X' = [ I_x' , I_x'y' ]
Y' = [ I_y' , -I_x'y' ]


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