Final answer:
In the center of mass frame, angular momentum conservation implies that the angular momentum of individual particles in a two-body system is proportional to the total angular momentum. The relationship can be expressed for each particle in terms of its mass relative to the total mass of the system.
Step-by-step explanation:
The question addresses the concept of angular momentum conservation in the center of mass (CM) frame. In physics, it is a fundamental principle that angular momentum is conserved in a system where net external torque is absent. This principle implies that the total angular momentum before and after any event (such as a collision) remains the same. Here, we are to show that the angular momentum of individual particles in a two-body system, within the CM frame, is directly proportional to the total angular momentum of the system.
To show this, let's consider a two-particle system with masses m_1 and m_2 and total mass M = m_1 + m_2. The CM frame is such that the velocity of the center of mass is zero. In this frame, the angular momentum ℓ_1 of particle 1 with respect to the center of mass is related to its mass and the total angular momentum L. The expression ℓ_1 = (m_2/M)L indicates that particle 1's angular momentum is proportional to the ratio of the mass of particle 2 to the total mass of the system, multiplied by the total angular momentum. Similarly, the angular momentum ℓ_2 for particle 2 is given by ℓ_2 = (m_1/M)L, showing an analogous proportionality.