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 the same whatever be the point about which the angular momentum is taken.

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We have two particles, each with mass mm and speed vv, traveling in opposite directions along parallel lines separated by a distance dd. To demonstrate that the vector angular momentum of this system remains the same regardless of the chosen point, let's denote the position vectors of the two particles as r1r1 and r2r2, respectively.

The angular momentum LL of a particle about a point OO is given by the cross product of its position vector rr and its linear momentum pp relative to that point:

LO=r×pLO=r×p

Now, let's calculate the angular momentum of each particle about an arbitrary point OO along their paths. For particle 1:

L1O=r1×p1L1O=r1×p1

And for particle 2:

L2O=r2×p2L2O=r2×p2

Since the particles are moving along parallel lines, their position vectors are parallel, and their magnitudes are equal, but their directions are opposite. Hence, r1=−r2r1=−r2.

Now, let's analyze the angular momentum of the system about point OO. The total angular momentum LtotalLtotal is the sum of the individual angular momenta:

Ltotal=L1O+L2OLtotal=L1O+L2O

Substituting the expressions for L1OL1O and L2OL2O, we get:

Ltotal=(r1×p1)+(r2×p2)Ltotal=(r1×p1)+(r2×p2)

Since r1=−r2r1=−r2, we can rewrite this as:

Ltotal=r1×(p1−p2)Ltotal=r1×(p1−p2)

Now, p1=mvp1=mv and p2=−mvp2=−mv (as the particles are moving in opposite directions), so:

p1−p2=mv−(−mv)=2mvp1−p2=mv−(−mv)=2mv

Substituting this back into the equation:

Ltotal=r1×(2mv)Ltotal=r1×(2mv)

Since r1r1 is the position vector of particle 1 relative to point OO, it doesn't change when we choose a different point. Therefore, LtotalLtotal remains the same regardless of the chosen point about which the angular momentum is calculated.

This demonstrates that the vector angular momentum of the two-particle system remains constant irrespective of the reference point chosen.