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Systems of Particles and Rotational Motion

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Question : 7
Total: 33
Two particles, each of mass m and speed v, travel in opposite directions along parallel lines separated by a distance d. Show that the vector angular momentum of the two particle system is the same whatever be the point about which the angular momentum is taken.
Solution:  
As shown in figure given below, suppose the two particles move parallel to the y-axis
Total angular momentum of the two particle system about O is
→
l
=
→
l
1+
→
l
2=
→
r
1×
→
p
1+
→
r
2×
→
p
2
=x
^
i
×(−mv)
^
j
+(x+d)
^
i
×(mv)
^
j
=(−mvx)
^
i
×
^
j
+(mvx+mvd)
^
i
×
^
j
=(−mvx+mvx−mvd)
^
k

=mvd
^
k
[∵
^
i
×
^
j
=
^
k
]
Clearly,
→
l
 does not depend on x and hence on the origin O. Thus the angular momentum of the two particle system is same whatever be point about which the angular momentum is taken.
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