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Sahithyan's S1
Sahithyan's S1 — Mechanics

Statics

Centroid

Aka. centre of area, geometric centre. The point where the area of a figure is assumed to be concentrated. Defined for a 2D shape.

Located on the axes of symmetry, if one exists.

Center of gravity

The point where the mass of an object is concentrated. Defined for a 3D shape.

First moment of area

Measure of spatial distribution of a shape in relation to an axis.

About x-axis=Ay  dA=Axˉ\text{About x-axis} = \int_A {y\; \text{d}A} = A\bar{x} About y-axis=Ax  dA=Ayˉ\text{About y-axis} = \int_A {x\;\text{d}A} = A\bar{y}

Here:

  • xˉ\bar{x} - Centroid’s xx coordinate
  • yˉ\bar{y} - Centroid’s yy coordinate
  • AA - Total area

About an axis of symmetry, first moment of area is 00.

Second moment of area

Aka. area moment of inertia. Describes how the cross-sectional area of a shape is distributed relative to an axis.

Not to be confused with moment of inertia.

About x-axis=Ixx=Ix=Ay2  dA\text{About x-axis} = I_{xx} = I_x = \int_A {y^2\;\text{d}A} About y-axis=Iyy=Iy=Ax2  dA\text{About y-axis} = I_{yy} = I_y = \int_A {x^2\;\text{d}A}

Always positive.

For common shapes

ShapeDescriptionIxxI_{xx}
Rectangle or ParallelogramBase bb. Height hh. About centroidal axis parallel to base.bh312\cfrac{bh^3}{12}
TriangleBase bb. Height hh. About base.bh312\cfrac{bh^3}{12}
TriangleBase bb. Height hh. About centroidal axis parallel to base.bh336\cfrac{bh^3}{36}
CircleDiameter dd. About centroidal axis.πd464\cfrac{\pi d^4}{64}

The product of moment of area about x,y axes

Ixy=Axy  dAI_{xy} = \int_A {xy\; \text{d}A}

The polar moment of area about z axis

Izz=J0=Ar2  dA=Ixx+IyyI_{zz} = J_0 = \int_A {r^2\; \text{d}A} = I_{xx} + I_{yy}

Radius of gyration

About x-axis=rx2=IxxA\text{About x-axis} = r_x^2 = \frac{I_{xx}}{A} About y-axis=ry2=IyyA\text{About y-axis} = r_y^2 = \frac{I_{yy}}{A} About z-axis=rz2=IzzA\text{About z-axis} = r_z^2 = \frac{I_{zz}}{A}

Moment of inertia

Describes the resistance of a body to changes in its rotational motion.