The Angular Momentum of a System of Particles — Question 5

Question 5

An impulsive couple changes internal motion without center recoil

Two particles at $\vec r_1=(1,0,0)$ and $\vec r_2=(-1,0,0)\,\mathrm m$ receive simultaneous impulses $\vec J_1=(0,2,0)$ and $\vec J_2=(0,-2,0)\,\mathrm{N\,s}$. Positions do not change appreciably during the impulses.

1. Find changes in total linear and angular momentum.

2. Explain why the center velocity remains unchanged for any positive particle masses.

Question 5 - Solution

An impulsive couple changes internal motion without center recoil

(a) Derivation and calculation

The total impulse is $$\boxed{\Delta\vec P=\vec J_1+\vec J_2=2\hat y-2\hat y=\vec0}.$$ The angular impulse is $$\Delta\vec L=(\hat x)\times(2\hat y)+(-\hat x)\times(-2\hat y) =\boxed{4\hat z\ \mathrm{kg\,m^2/s}}.$$

(b) Result and interpretation

For any positive total mass, $$M\Delta\vec V_C=\Delta\vec P=0\Rightarrow\boxed{\Delta\vec V_C=0}.$$ Positions do not appreciably change during the pulse, so the center orbital contribution remains fixed. The angular impulse changes internal angular momentum while leaving the center velocity unchanged.

Physical check

The two impulses have opposite linear effects but the same signed angular effect.