Question 7 of 16advanced🔍 AnalyzeLong Answer3 marks

Two particles, each of mass m and speed v, travel in opposite directions along parallel lines separated by a distance d. Show that the angular momentum vector of the two particle system is the same whatever be the point about which the angular momentum is taken.

Correct Answer

Let the two particles be P₁ and P₂, each of mass m and speed v, moving in opposite directions along parallel lines separated by distance d.

Let us calculate the angular momentum about an arbitrary point O. The angular momentum of the system is given by L = l₁ + l₂ = r₁ × p₁ + r₂ × p₂, where r₁ and r₂ are position vectors of the two particles relative to O, and p₁ = mv₁, p₂ = mv₂ are their linear momenta.

Now, if we shift the origin to another point O', the new position vectors become r₁' = r₁ - d and r₂' = r₂ - d, where d is the displacement vector from O to O'. The angular momentum about O' becomes:

L' = r₁' × p₁ + r₂' × p₂

= (r₁ - d) × p₁ + (r₂ - d) × p₂

= r₁ × p₁ + r₂ × p₂ - d × (p₁ + p₂)

= L - d × (p₁ + p₂)

Since the particles have equal mass m and equal speed v but move in opposite directions, their velocities are v and -v respectively. Hence, p₁ + p₂ = mv + m(-v) = 0.

Therefore, L' = L - d × 0 = L.

This proves that the angular momentum vector of the two-particle system remains the same whatever be the point about which it is calculated. This occurs because the total linear momentum of the system is zero.

Exercise: EXERCISES | Q: 6.7 | (Page 34)
For More Understanding

Explanation

The question tests understanding of angular momentum for a system of particles. The key insight is that when total linear momentum of a system is zero, the angular momentum becomes independent of the choice of origin. Since the two particles have equal masses and speeds but opposite velocities, their linear momenta cancel out (p₁ + p₂ = 0). Using the relation L' = L - d × P_total from the context, where P_total = 0, we get L' = L. The angular momentum has constant magnitude mvd (perpendicular to the plane of motion) and fixed direction.

Solution Steps

  1. Step 1: Express total angular momentum L = r₁ × p₁ + r₂ × p₂ about point O

  2. Step 2: For a new origin O', write new position vectors r₁' = r₁ - d and r₂' = r₂ - d

  3. Step 3: Calculate L' = (r₁ - d) × p₁ + (r₂ - d) × p₂ = L - d × (p₁ + p₂)

  4. Step 4: Show that p₁ + p₂ = 0 since particles have equal and opposite velocities

  5. Step 5: Conclude L' = L, proving angular momentum is independent of origin choice

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