Question

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 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.

Answer 1

l_x = yp_z – zp_y

l_y = zp_x – xp_z

l_z = xp_y –yp_x

Linear momentum of the particle,

Position vector of the particle,

Angular momentum,

Comparing the coefficients of we get:

The particle moves in the x-y plane. Hence, the z-component of the position vector and linear momentum vector becomes zero, i.e.,

z = p_z = 0

Thus, equation (i) reduces to:

Therefore, when the particle is confined to move in the x-y plane, the direction of angular momentum is along the z-direction.