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# Find the components along the x , y, z axes of the angular momentum l of a particle whose position vector is r with components x , y, z.

Q7.6  Find the components along the $x,y,z$  axes of the angular momentum $1$  of a particle, whose position vector is $r$ with components $x,y,z$  and momentum is  $p$ with components$p_{x}$,$p_{y}$ and $p_{z}$ . Show that if the particle moves only in the $x-y$ plane the angular momentum has only a z-component.

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Linear momentum of particle is given by :

$\overrightarrow{p}\ =\ p_x \widehat{i}\ +\ p_y \widehat{j}\ +\ p_z \widehat{k}$

And the angular momentum is :

$\overrightarrow{l}\ = \overrightarrow{r}\times \overrightarrow{p}$

$= \left ( x \widehat{i}\ +\ y \widehat{j}\ +\ z \widehat{k} \right ) \times \left ( p_x \widehat{i}\ +\ p_y \widehat{j}\ +\ p_z \widehat{k} \right )$

$=\begin{vmatrix} i &j &k \\ x &y &z \\ P_x &P_y &P_z \end{vmatrix}$

$=\ \widehat{i}\left ( yp_z - zp_y \right )\ -\ \widehat{j}\left ( xp_z - zp_x \right )\ +\ \widehat{k}\left ( xp_y - yp_x \right )$

When particle is confined to x-y plane then z = 0 and  pz = 0.

When we put the value of z and pz in the equation of linear momentum then we observe that only the z component is non-zero.

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