System of particles and Rotational motion NCERT Solutions Class 11 Physics - Solved Exercise Question 7.6

Question 7.6:
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 px, py and pz. Show that if the particle moves only in the x-y plane the angular momentum has only a z-component.
Solution:
lx = ypzzpy
ly = zpx – xpz
lz = xpyypx
Linear momentum of the particle, p = px i + py j + pz k
Position vector of the particle, r = x i + y j + z k
Angular momentum, l = r X p

= (x i + y j + z k) X (px i + py j + pz k)

=   i       j       k
     x      y       z
    px     py     pz

lx i + ly j + lz k = i (ypz - zpy) - j(xpz - zpx) + k(xpy - zpx)
Comparing the coefficients of i, j and k, we get:
lx = ypz - zpy
ly = xpz - zpx
lz = xpy - ypx
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 = pz = 0
Thus, equation (i) reduces to:
lx = 0
ly = 0
lz = xpy - ypx
Therefore, when the particle is confined to move in the x-y plane, the direction of angular momentum is along the z-direction.

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