Test Index

Systems of Particles and Rotational Motion

© examsnet.com
Question : 6
Total: 33
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,py‌and‌pz. Show that if the particle moves only in the x-y plane the angular momentum has only a z-component.
Solution:  
(a) Angular momentum,
→
L
=
→
r
×
→
p
It is a vector quantity and its direction is given by right hand rule for vector product. As
→
r
 and
→
p
 here lie in xoy plane, so
→
L
 acts along Z axis.
In cartesian co-ordinates,
and
→
r
=x
^
i
+y
^
j
+z
^
k
→
p
=px
^
i
+py
^
j
+pz
^
k
}...(ii)
∴ From (i) and (ii), we get
→
L
=(x
^
i
+y
^
j
+z
^
k
)×(px
^
i
+py
^
j
+pz
^
k
)
=|
^
i
^
j
^
k
xyz
pxpypz
|
or Lx
^
i
+Ly
^
j
+Lz
^
k
=
^
i
(ypz−zpy)+
^
j
(zpx−xpz)+
^
k
(xpy−ypx)
On comparing, we get
Lx=ypz−zpy
Ly=zpx−xpz
Lz=xpy−ypx
}...(iii)
Equation (iii) gives the required components of L along x,y and z axes.
(b) As the particle moves in x−y plane, then
z=0 and pz=0
Hence,
→
L
=
^
k
(xpy−ypx)=Lz
^
k

Hence angular momentum has only z-component.
© examsnet.com
Go to Question:
Prev Question
Next Question