Two particles, each of mass m and speed v, ...

Systems of Particles and Rotational Motion

Question : 7 of 33

Marks: +1, -0

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

\vec{l} = \vec{l}_1 + \vec{l}_2 = \vec{r}_1 \times \vec{p}_1 + \vec{r}_2 \times \vec{p}_2

= x \hat{i} \times (-m v) \hat{j} + (x+d) \hat{i} \times (m v) \hat{j}

= (-m v x) \hat{i} \times \hat{j} + (m v x + m v d) \hat{i} \times \hat{j}

= (-m v x + m v x - m v d) \hat{k}

= m v d \hat{k} [\because \hat{i} \times \hat{j} = \hat{k}]

Clearly,

\vec{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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