The Angular Momentum of a System of Particles — Question 9

Question 9

Opposite angular momenta can coexist with substantial kinetic energy

Two $1\,\mathrm{kg}$ particles are at $(1,0,0)$ and $(-1,0,0)\,\mathrm m$ and both move at $2\hat y\,\mathrm{m/s}$.

1. Find total angular momentum about the origin and total kinetic energy.

2. Describe their center-frame motion and explain the cancellation.

Question 9 - Solution

Opposite angular momenta can coexist with substantial kinetic energy

(a) Derivation and calculation

The two angular momenta cancel: $$\vec L_1=\hat x\times2\hat y=2\hat z,\qquad \vec L_2=(-\hat x)\times2\hat y=-2\hat z,$$ $$\boxed{\vec L_O=\vec0}.$$ Their positive kinetic energies add: $$\boxed{K=2\left[\tfrac12(1)(2)^2\right]=4.00\ \mathrm J}.$$

(b) Result and interpretation

The center is at the origin and has velocity $\vec V_C=2\hat y$. Each relative velocity is $$\vec u_i=\vec v_i-\vec V_C=0.$$ Thus the motion is instantaneously pure translation, with $L_C=0$ and $\vec R_C\times M\vec V_C=0$. Signed angular momenta can cancel while scalar kinetic energies remain positive.

Physical check

The center translational energy $MV_C^2/2=2(4)/2=4$ J accounts for the entire energy.