question_answer

A particle of mass m moves in the XY plane with a velocity v along the straight line AB. If the angular momentum of the particle with respect to origin O is \[{{L}_{A}}\] when it is at A and \[{{L}_{B}}\] when it is at B, then

A) \[{{L}_{A}}>{{L}_{B}}\]              

B) \[{{L}_{A}}={{L}_{B}}\]

C) the relationship between \[{{L}_{A}}\,and\,\,{{L}_{B}}\] depends upon the slope of the line AB

D) \[{{L}_{A}}\,<\,\,{{L}_{B}}\]

Correct Answer: B

Solution :

From the definition of angular momentum, \[\vec{L}= \vec{r} \times \,\,\vec{P}\,\,=\,\,rm\nu \,\,sin\,\,\phi \,\,\left( -\hat{k} \right)\] Therefore, the magnitude of L is \[\operatorname{L} = mvr\,\,sin \phi \,\,=\,\,mvd\] where \[\operatorname{d}= r\,\,sin\phi \] is the distance of closest approach of the particle to the origin. As d is the same for both the particles, hence \[{{L}_{A}}={{L}_{B}}\].