The Angular Momentum of a System of Particles — Question 8

Question 8

Fragment spin belongs in the angular momentum budget

An isolated object initially has angular momentum $8\hat z\,\mathrm{kg\,m^2/s}$ about its stationary mass center. It breaks into two extended fragments. Their center motions carry orbital angular momenta $3\hat z$ and $-1\hat z$ in the same units.

1. Find the sum of their intrinsic spin angular momenta.

2. If fragment 1 has spin $4\hat z$, find fragment 2 spin and explain the error in treating both as structureless particles.

Question 8 - Solution

Fragment spin belongs in the angular momentum budget

(a) Derivation and calculation

Conservation includes both fragment-center motion and intrinsic spin: $$\vec L_i=\vec L_{{\mathrm{orb}},1}+\vec L_{{\mathrm{orb}},2}+\vec S_1+\vec S_2.$$ Thus $$8\hat z=3\hat z-\hat z+\vec S_1+\vec S_2,$$ $$\boxed{\vec S_1+\vec S_2=6\hat z\ \mathrm{kg\,m^2/s}}.$$

(b) Result and interpretation

If $\vec S_1=4\hat z$, then $$\boxed{\vec S_2=(6-4)\hat z=2\hat z\ \mathrm{kg\,m^2/s}}.$$ Treating both fragments as structureless particles would retain only $3-1=2$ of the original 8 units. The missing 6 units reside in fragment spin, so that simplified model would omit essential angular momentum.

Physical check

The complete accounting is $3-1+4+2=8$, despite any changes in kinetic energy during breakup.