The Linear Momentum of a System of Particles — Question 8

Question 8

A region is not always a fixed system

Two noninteracting $1\,\mathrm{kg}$ beads move rightward at $2\,\mathrm{m/s}$. A camera reports momentum only for beads inside $0<x<1\,\mathrm m$. Initially one bead is inside. During the next interval it exits, and the other bead remains outside.

1. Find the reported momentum change for the region and the momentum change of the two-bead system.

2. Does the regional change prove that a force acted? State the boundary issue.

Question 8 - Solution

A region is not always a fixed system

(a) Derivation and calculation

The camera counts a changing set of particles: $$P_{{\mathrm{region}},i}=1(2)=2,\qquad P_{{\mathrm{region}},f}=0, \quad\boxed{\Delta P_{\mathrm{region}}=-2\ \mathrm{kg\,m/s}}.$$ For the fixed two-bead system, $$P_{{\mathrm{system}},i}=P_{{\mathrm{system}},f}=2(1)(2)=4, \qquad\boxed{\Delta P_{\mathrm{system}}=0}.$$

(b) Result and interpretation

The regional balance includes transport across the boundary: $$\Delta P_{\mathrm{region}}=J_{\mathrm{ext}}+P_{\mathrm{in}}-P_{\mathrm{out}}=0+0-2.$$ No force is inferred from this loss: the exiting bead carries its unchanged momentum out of the observed region.

Physical check

The fixed-particle momentum remains constant even though the camera no longer counts one contribution.