Components of a Force

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Forces acting at some angle from the the coordinate axes can be resolved into mutually perpendicular forces called components. The component of a force parallel to the x-axis is called the x-component, parallel to y-axis the y-component, and so on.
 

Components of a Force in XY Plane
008-components-force-xy-plane-2d.jpg$ F_x = F \cos \theta_x = F \sin \theta_y $

$ F_y = F \sin \theta_x = F \cos \theta_y $

$ F= \sqrt{{F_x}^2 + {F_y}^2} $

$ \tan \theta_x = \dfrac{F_y}{F_x} $
 

009-components-force-given-slope.jpgGiven the slope of the line of action of the force as v/h (see figure to the right)
$ r =\sqrt{h^2 + v^2} $

$ F_x = F (h/r) $

$ F_y = F (v/r) $
 

Components of a Force in 3D Space
010-components-3d-force-direction-cosines.jpgGiven the direction cosines of the force:
$ F_x = F \cos \theta_x $

$ F_y = F \cos \theta_y $

$ F_z = F \cos \theta_z $

$ F = \sqrt{{F_x}^2 + {F_y}^2 + {F_z}^2} $

$ \cos \theta_x = \dfrac{F_x}{F} $

$ \cos \theta_y = \dfrac{F_y}{F} $

$ \cos \theta_z = \dfrac{F_z}{F} $
 

Components of a force in 3D spaceGiven the coordinates of any two points along the line of action of the force (in reference to the figure shown, one of the points is the origin):
Let d = distance OB
$ d = \sqrt{x^2 + y^2 + z^2} $

$ F_x = F (x/d) $

$ F_y = F (y/d) $

$ F_z = F (z/d) $
 

 

Vector Notation of a Force (Also called Rectangular Representation of a Force)

$ {\bf F} = F{\bf \lambda} $

Where λ is a unit vector. There are two cases in determining λ; by direction cosines and by the coordinates of any two points on the line of action of the force.
 

Given the direction cosines:
$ {\bf \lambda} = \cos \theta_x {\bf i} + \cos \theta_y {\bf j} + \cos \theta_z {\bf k} $

Given any two points P1(x1, y1) and P2(x2, y2) on the line of action of the force:
$ {\bf \lambda} = \dfrac{1}{d} \left( d_x{\bf i} + d_y {\bf j} + d_z {\bf k} \right) $

Where
i, j, and k are unit vectors in the direction of x, y and z respectively.
$ d_x = x_2 - x_1 $
$ d_y = y_2 - y_1 $
$ d_z = z_2 - z_1 $

$ d = \sqrt{{d_x}^2 + {d_y}^2 + {d_z}^2} $

$ \cos \theta_x = d_x / d $
$ \cos \theta_y = d_y / d $
$ \cos \theta_z = d_z / d $

Note:
$ \cos^2 \theta_x + \cos^2 \theta_y + \cos^2 \theta_z = 1 $
$ (d_x / d)^2 + (d_y / d)^2 + (d_z / d)^2 = 1 $

Also note the following:
$ F_x = \cos \theta_x = d_x / d $
$ F_y = \cos \theta_y = d_y / d $
$ F_z = \cos \theta_z = d_z / d $

$ F = \sqrt{{F_x}^2 + {F_y}^2 + {F_z}^2} $
 

Thus,

$ {\bf F} = F ( \cos \theta_x {\bf i} + \cos \theta_y {\bf j} + \cos \theta_z {\bf k}) $

$ {\bf F} = \dfrac{F}{d} \left( d_x{\bf i} + d_y {\bf j} + d_z {\bf k} \right) $

 

In simplest term

$ {\bf F} = F_x {\bf i} + F_y{\bf j} + F_z{\bf k} $

 

The above rectangular representation of a force is applicable in both 2D and 3D forces.
 

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Comments

engineering mechanics

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Re: Components of a Force

Hello,
Can you please check the formulas once again. I may be wrong, but there is some error in the Angle consideration.

Thanks for uploading, it was useful.

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