User:Duneatreides/moment

User:Duneatreides/moment(physics)

In physics, the moment of a force, is a quantity that measures the tendency of a force to cause a body to rotate. The moment of a force is often shorten to just moment and the moment of a force is also known as Torque. The SI unit for moment is the newton meter N\middot\m. The moment is a vector quantity.

Scalar Formation
The magnitude of the moment is $$ \vert \vec M_o \vert = \vert \vec F \vert d_\perp$$

where

$$ \vert \vec M_o \vert $$ is the magnitude of moment about the o axis.

$$ \vert \vec F \vert $$ is the magnitude of the force.

$$ d_\perp $$ is the perpendicular distance to the line of action of $$ \vert \vec F \vert $$.

The quantity $$ d_\perp $$ is also called the moment arm or lever arm.

Vector Formation
The scalar formation is only useful when one can easily find $$d_\perp$$, however in most cases finding the moment arm is extremely challenging. It is more efficient to use the cross product.

The moment is defined as $$ \vec M_o = \vec r \times \vec F $$.

Calculating The Momemnt (Vector Analysis)
Let

$$ \vec r = r_x \vec i + r_y \vec j + r_z \vec k $$

$$ \vec F = F_x \vec i + F_y \vec j + F_z \vec k $$

Then

$$\vec M_o = \vec r \times \vec F = \begin{vmatrix} \vec i & \vec j & \vec k \\ r_x & r_y & r_z \\ F_x & F_y & F_z \end{vmatrix} = \begin{vmatrix} r_y & r_z \\ F_y & F_z \end{vmatrix} \vec i - \begin{vmatrix} r_x & r_z \\ F_x & F_z \end{vmatrix} \vec j + \begin{vmatrix} r_x & r_y \\ F_x & F_y \end{vmatrix} \vec k $$

$$= \left [ \left (r_yF_z \right) - \left (r_zF_y \right) \right] \vec i - \left [ \left (r_xF_z \right) - \left (r_zF_x \right) \right] \vec j + \left [ \left (r_xF_y \right) - \left (r_yF_x \right) \right] \vec k $$

$$ \left | \vec M_o \right \vert = \left | \vec r \times \vec F \right \vert = \left | \vec r \right \vert \left | \vec F \right \vert \sin \theta $$