Two identical particles have equal but opposite momenta, p→\overrightarrow{p} and −p→-\overrightarrow{p}, but they are not traveling along the same line. Show that the total angular momentum of this system does not depend on the choice of origin.

Ch. 11 - Angular Momentum; General Rotation

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

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Giancoli Douglas 5th edition

Ch. 11 - Angular Momentum; General Rotation

문제 37

11장, 문제 37

Two identical particles have equal but opposite momenta, $\vec{p}$ and $- \vec{p}$ , but they are not traveling along the same line. Show that the total angular momentum of this system does not depend on the choice of origin.

검증된 단계별 안내

Start by recalling the definition of angular momentum for a particle. The angular momentum of a particle relative to a chosen origin is given by the vector cross product:

L⃗ = r⃗ × p⃗

, where

r⃗

is the position vector of the particle relative to the origin, and

p⃗

is its linear momentum.

For the system of two particles, the total angular momentum is the vector sum of the angular momenta of the two particles:

L⃗ _{total} = L⃗ _1 + L⃗ _2 = (r⃗ _1 × p⃗ _1) + (r⃗ _2 × p⃗ _2)

, where subscripts 1 and 2 refer to the two particles.

Substitute the given momenta into the equation. The first particle has momentum

p⃗ _1 = p⃗

, and the second particle has momentum

p⃗ _2 = -p⃗

. Thus, the total angular momentum becomes:

L⃗ _{total} = (r⃗ _1 × p⃗) + (r⃗ _2 × (-p⃗))

Simplify the expression by factoring out

p⃗

L⃗ _{total} = (r⃗ _1 - r⃗ _2) × p⃗

. Notice that the total angular momentum depends only on the relative position vector

r⃗ _1 - r⃗ _2

and the momentum

p⃗

, not on the choice of origin.

Conclude that since the relative position vector

r⃗ _1 - r⃗ _2

is independent of the origin, the total angular momentum

L⃗ _{total}

does not depend on the choice of origin. This completes the proof.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Angular Momentum

Angular momentum is a vector quantity that represents the rotational motion of an object. It is defined as the product of the position vector and the momentum of the particle. For a system of particles, the total angular momentum is the vector sum of the angular momenta of each particle, which depends on both their momenta and their positions relative to a chosen origin.

Conservation of Angular Momentum

The conservation of angular momentum states that if no external torque acts on a system, the total angular momentum remains constant. This principle is crucial in analyzing systems of particles, as it implies that the angular momentum calculated from different origins will yield the same total value, provided the internal interactions are unchanged.

Choice of Origin

The choice of origin in a coordinate system can affect the calculation of position vectors but does not affect the physical laws governing the system. When analyzing angular momentum, shifting the origin will change the position vectors of the particles, but the total angular momentum remains invariant due to the symmetrical nature of the momenta involved, particularly when they are equal and opposite.